CHAPTER ONE

INTRODUCTION

At the national level, the grassroots level, and all levels in between, a dialogue is

underway regarding directions for mathematics education. The nature of curriculum, the

roles of teacher and learner, and appropriate uses of technology are but a few subjects of

lively debate. A great many sacred cows are in danger of being slaughtered; one wonders

which shall be spared. Yet the more things change, as they say, the more they stay the

same. When discussing educational matters, this has particular weight, considering that

the core operations of schooling have changed little since the nineteenth century (Tyack

& Cuban, 1995).

Still, this could be a watershed moment in mathematics education. Perhaps this

point in history is a nexus of critical events including, but not limited to: advances in

technology, developments in cognitive and social psychology, the emergence of highly-

publicized international studies of school achievement, the acknowledgement of

educational inequities, and the purported intentions to address these inequities, however

sporadically. In whatever light history casts on these events, the door to a different

mathematics education is open; who passes through it remains to be seen.

Of course, one might argue that any real change must happen from the ground up.

That is, the determination whether the status quo is rejected or reaffirmed will be made in

the not-so-routine daily decisions of the classroom teacher in the context of a particular

mathematics department (D'Ambrosio et al, 1992). As Fullan and Stiegelbauer (1991)

suggest, "Educational change depends on what teachers do and think-it's as simple and as

complex as that" (p. 117).

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Although issuing more government reports or state requirements could lead to

some surface change, to a teacher these acts may only represent hollow political

posturing, lacking real transformative power in the classroom. Such outside mandates

may be treated as simply another educational fad which will pass in time, as have so

many others.

For example, the Mathematical Sciences Education Board has suggested that

school mathematics must be equitable, and should not filter students out of scientific or

professional careers (1989). Yet, while Volmink (1994) agrees that such filtering is

undesirable, he argues that this has in fact been the historical function of mathematics

education. He writes, "Mathematics is not only an impenetrable mystery to many, but has

also, more than any other subject, been cast in the role as an 'objective' judge, in order to

decide who in society 'can' and who 'cannot'" (p. 51).

So, the rejection of this gatekeeping function of school mathematics, as well as

the acceptance of other "reform-oriented" recommendations, would indicate a significant

shift in philosophy by the mathematics education community. Certainly, this sort of shift

could not be mandated for the profession by the state, for the principles underlying

different models of school mathematics are deeply rooted, and beliefs are often

passionately held by educators. Instead, such a philosophical transformation would

require an intense period of debate and soul-searching by the profession as a whole

Again, the present time may be such a period.

When such a transformation occurs for an individual teacher, it might be likened

to a professional “Copernican Revolution”, a complete paradigm shift regarding the

fundamental assumptions of mathematics education. Whether this shift is a painstaking

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evolution or a sudden awakening, the discovery that one’s profession has been spinning

on a different axis may be simultaneously exciting, intimidating, and perhaps frightening.

As Norum and Lowry (1995) suggest, "When a change occurs, for some, it will be

uncomfortable but manageable. For others, it may be downright terrifying" (p. 4).

Statement of the Problem

Although the process of change is uncomfortable, mathematics departments may

be forced to reexamine the traditional habits, methods, and attitudes that have guided

practice for many years, and which may no longer be appropriate for a changing

American populace. Yet, as Gutierrez (1996) suggests, if we are to improve opportunities

for all students in mathematics education, we need to examine the learning environments

of students who traditionally underperform in mathematics. This paper is a snapshot of a

mathematics department in a large, comprehensive, suburban public high school that has

begun such an effort, and has actively undertaken steps to move toward a reformed vision

of mathematics education.

Specifically, the mathematics department at Adderley High School (a pseudonym)

has begun to phase out a "traditional" mathematics sequence in favor of the Core-Plus

Mathematics Project (CPMP), a curriculum designed to encourage teaching consistent

with such documents as the National Council of Teachers of Mathematics (NCTM)

Curriculum and Evaluation Standards (1989), and Principles and Standards for School

Mathematics (2000). As a result of the curriculum change, and turnover in teaching staff

and district leadership, the department has been in a period of substantial restructuring

since1998. That is, this mathematics department has been forced to critically analyze its

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instructional practices, methods of student placement, use of technology, and professional

relationships. In short, they are operating in a new paradigm.

In this paper, I discuss the successes and challenges experienced by this

mathematics department in transition, and I analyze some cultural traits that have

facilitated the department's receptivity to this educational paradigm shift.

Explanation of Terminology

It is essential to clarify what is meant by "reform-oriented" versus "traditional"

mathematics instruction in this paper. Many reforms of mathematics education have been

proposed, but the most appropriate to this discussion are those described in the NCTM

Curriculum and Evaluation Standards (1989), Professional Standards for Teaching

(1991), and Principles and Standards for School Mathematics (2000). However, I do not

wish to suggest that "traditional" instruction and "reform-oriented" teaching are mutually

exclusive. Yet, compared to "traditional" teaching, a "reform-oriented" vision of

mathematics education is supported by very different principles and generally employs

very different methods. These underlying differences are fundamental and should be

made explicit.

Explanation of "Reform-Oriented Mathematics"

Some crucial differences between reform-oriented mathematics and traditional

mathematics are found in the roles envisioned for both the teacher and the students. In a

reform-oriented mathematics paradigm, students are both actively and passively involved

in mathematical problem-solving and application of mathematical ideas (National

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Council of Teachers of Mathematics, 1989). The teacher's role, therefore, is more of a

guide or facilitator of student activity rather than a lecturer. Of course, there will

necessarily be some presentation of material, but the use of lecture as an instructional

strategy is certainly de-emphasized (D'Ambrosio et al, 1992; National Council of

Teachers of Mathematics, 1989). In addition to these roles, other hallmarks of reform-

oriented mathematics include:

Increased student interaction and classroom discourse The use of a variety of instructional techniques, such as cooperative learning

and projects, as well as individual work Activities arising out of problem situations The position that all students should have access to learning significant

mathematics, as opposed to a select few Heterogeneous student grouping Increased use of appropriate technology, such as calculators or computers Covering fewer mathematical topics at greater depth Employing multiple methods of assessment

( Mathematical Sciences Education Board, 1989; Mathematical Sciences Education

Board, 1990; National Council of Teachers of Mathematics, 1989; National Council of

Teachers of Mathematics, 1991; National Council of Teachers of Mathematics, 1998).

These suggestions stem in part from important research in cognitive development, the

need to meet the demands of an information society, and the duty to provide an adequate,

equitable education for all students.

Explanation of "Traditional Mathematics"

In contrast, "traditional" American mathematics instruction has been characterized

by a familiar daily routine: the teacher checks the answers to the previous day's

homework, works problems on the board, introduces a new concept, works some

examples, and assigns seatwork (Cobb, 1992; Stigler & Hiebert, 1999). In contrast to

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reform recommendations, teacher exposition is the primary method of instruction, and the

student has the passive role of listening and taking notes. Furthermore, rote

memorization of facts and procedures dominates instruction, characterized by paper-and

pencil skill work (National Council of Teachers of Mathematics, 1989).

In addition, traditional mathematics has been characterized as mathematics-for-

the-elite, and significant mathematics is taught only to the top-performing students

(D'Ambrosio et al, 1992; Volmink, 1994). Therefore, sorting students into ability groups,

or "tracking", is a common practice.

Volmink (1994) presents this rather pessimistic view of traditional mathematics:

There are strong hegemonic forces in our society, that impose a certain view of mathematics on us all. Our schooling in many ways has encouraged us to accept as unproblematic, that the traditional mathematics curriculum somehow embodies uniquely powerful knowledge and eternal truths which should be taught and learned in a catechistic fashion. Furthermore, this draconian body of knowledge is not only infallible but also universal. (p. 52)

Stigler and Hiebert (1999) found that mathematics instruction is far from

"universal"; Japanese, German, and American teachers differ drastically in their

assumptions, methods, and goals. Certainly, these strong words from Volmink

reflect the passion that mathematics education can inspire. I have deliberately

drawn stark distinctions between reform and traditional mathematics, but this is

not to inflame emotions or to oversimplify the choices teachers make. Classroom

teaching is rarely black-and-white; most teachers probably operate in shades of

gray. Yet, the distinctions help explicate the current situation unfolding at

Adderley High Schools.

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School Context

Adderley Township Community High Schools consist of two large public

institutions: East Adderley and West Adderley. These buildings serve approximately

1,700 and 1,500 students, respectively. The pupil-teacher ratio in the district is 18.8 to 1,

and there are 22 total mathematics teachers employed. Although several miles apart, both

schools are considered one unit. That is, students at East and West Adderley compete on

the same athletic teams, and the respective academic departments have one department

chair travelling between campuses. Still, each school has an administrative team

consisting of a principal, an assistant principal, and two deans of students.

Despite the fact that the district is in suburban Chicago, both schools are located

in urban environments. Industrial and retail areas have built up around the schools,

providing the district with a significant tax base. Despite being well-funded, the district

confronts many of the same problems facing urban schools. For example, a zero-

tolerance policy toward gang activity on school grounds is strictly enforced. To this end,

students may not wear clothing that bears the insignia or colors of certain popular sports

teams, as these are associated with gangs in the area. This is not to suggest that the

schools are dangerous or in some way chaotic. Student discipline is handled efficiently,

and the school grounds are well-kept and orderly.

Student Information

To provide further background, some student demographic information may be

illustrative. According to the Illinois School Report Card, the district has a sizable

Limited-English-Proficient (LEP) population. In fact, 12.8% of the student body is LEP,

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which is exactly double the Illinois state average. The student population is also racially

and ethnically diverse, although this is not uniformly distributed throughout the district

[See Table 1]. The district-wide average class size is between 21 and 22 students, slightly

above the Illinois State average of 18.3.

Table 1

Student Racial/Ethnic Background

School White Hispanic Black Asian/Pacific Islander

Native American

East Adderley H.S. 68.3% 26.7% 0.8% 3.6% 0.5%West Adderley H.S. 52.4% 40.7% 2.5% 4.3% 0.1%

Personal History with the District

I began my teaching career at West Adderley High School in 1996, after

graduating from the University of Illinois at Urbana-Champaign with a Bachelor's Degree

in the Teaching of Mathematics. At the time, the department had a traditionally

constructed curriculum that sorted students into "skills", "regular" and "honors" tracks.

Typically, a student would take a sequence of algebra as a freshman and geometry as a

sophomore. If this student decided to take mathematics beyond the district's two-year

mathematics requirement, some form of advanced algebra was generally next in the

sequence. Figure 1 illustrates this course sequence and structure.

While Algebra I and Geometry were offered at the "regular" level, Pre Algebra,

Algebra S, and Geometry S were remedial-level courses taken by the generally lower-

achieving "skills" students. Honors students started their high school mathematics

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education with Advanced Algebra Honors, and some proceeded as sophomores to

Geometry Honors Plus, a course team-taught with Honors Chemistry. From the figure

below, one notices a fairly stratified system. Yet it must be noted that this was not an

entirely rigid structure; students often moved between tracks with a teacher's

recommendation.

Grouping Freshman Sophomore Junior SeniorSkills Pre algebra Algebra S Geometry SRegular Algebra S or

Algebra IGeometry S or Geometry I

Algebra II College Algebra

Honors Advanced Algebra Honors

Geometry/Chemistry Honors Plus or Geometry Honors

Trig/PreCalc or Algebra II Honors

AP Calculus

Figure 1. Course selections, 1996-1997.

The seeds of this study were planted when a new department chair, Mr. Blakey (a

pseudonym), took over at the start of the 1997-1998 school year. He immediately

replaced the remedial Algebra courses (Algebra S) with the Core-Plus Mathematics

Program (CPMP) as a pilot program. I was one of six teachers in the department who

taught CPMP at this time. Of course, this was only the start of the changes that form the

basis of this paper.

I was personally intrigued by the reform-oriented perspective of mathematics

teaching and learning encouraged by the CPMP materials. Students who might ordinarily

have been disconnected from a traditional algebra course seemed interested in the Core-

Plus content because of the emphasis on presenting mathematics in a real-life context.

Many were able to transcend a lack of computational skill through the use of graphing

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calculator technology, a key feature of the curriculum. In addition, perhaps related to the

issue of computational skill, many told me that they were able to overcome a measure of

"mathematics anxiety", the fear of failure they had felt taking past courses. As a result, I

saw a great deal of potential in this program for all students. I believed that it would offer

opportunities for not only skills students, but also regular and honors students. That is,

they could all engage in real, meaningful mathematics while developing critical thinking

and problem solving skills. Details of the Core-Plus curriculum are discussed further in

Chapter 3.

After the 1997-1998 school year, the first year of the pilot program, I took a leave

of absence from Adderley. However, I was still interested in the program, which had

been successful enough to merit wider implementation. This research arose from keeping

contact with the department during my leave and graduate studies, and observing the

extent to which the Core-Plus curriculum was being phased into all classes and at all

levels. In my opinion, such a rapid and sweeping change in direction warranted further

examination.

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CHAPTER TWO

REVIEW OF LITERATURE

For this study, my focus has been to identify characteristics of a mathematics

department as a reform-oriented curriculum in the beginning stages of implementation.

As a result, I have sought documentation of the receptivity of mathematics teachers,

individually and collectively, to curricular and instructional change. That is, under what

conditions are teachers and departments ready for such a shift in direction?

Much has been documented on traits of teachers as a whole, and this is discussed

below. From this general investigation of teacher characteristics, the discussion moves to

research on indicators of mathematics teachers' receptivity toward reform-oriented

curricula and instructional methods. Specifically, this is research that notes teachers'

attitudes regarding the role of the teacher and instructional activities. Finally, I investigate

the characteristics, culture, and receptivity to change of groups of teachers: the

mathematics department as a whole.

Teacher Traits

It has been found that personality types are not distributed evenly among

professions (Lawrence, 1982). Certainly teaching is no exception. Research has shown

wide variation among teachers’ conceptions of control, motivation, self-actualization, and

desire for change (Ashton & Webb, 1986; Hopkins, 1990; Huberman, 1992; McKibbin &

Joyce, 1980; Rosenholtz, 1989). One scale often used to measure personality traits of

teachers is the Myers-Briggs Type Indicator (MBTI). Based on the work of C.G. Jung,

this inventory defines personality based on 16 types. Four bipolar scales are used in this

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measure: Extraversion-Introversion (EI), Sensation-Intuition (SN), Thinking-Feeling

(TF), and Judging-Perception (JP).

The EI scale measures one's tendency to obtain information through the external

world of people and things, or through the inner world of ideas. That is, extroverts are

more outgoing, and introverts tend to be more reflective. The SN scale reflects ways of

perceiving the world, either through the senses or by intuitive judgements. Sensation-

oriented individuals base their perceptions on real, concrete, objective data, as opposed to

the hunches and unconscious information that tend to guide Intuitive perceivers (Kent &

Fisher, 1997). The TF scale is a measure of the preferred method of judging experiences.

Experience may be judged mainly by logic (T), or by subjective, personal assessments

(F). Finally, JP refers to one's attitude toward the outside world. Judging types prefer to

have things decided according to an organized, rational plan. Perceiving types are

naturally prone to flexibility and spontaneity (Lawrence, 1982). Within each individual,

the four preferences interact to determine one's personality type.

For example, as measured on this scale, a person might be identified as more

predisposed to introversion over extroversion, intuition over sensation, feeling over

thinking, and perceiving over judging. Therefore, this person would be classified as

"INFP". However, some research has explored trends among subsets of these 16

personality types, specifically as they relate to teachers

For example, Lawrence notes that 67% of high school teachers are "judging"

types, whose classrooms are more likely to be orderly and governed according to

structure and schedules. This characterization is supported by Kent and Fisher (1997),

who suggest that judging type teachers see themselves as encouraging high levels of on-

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task behavior. In contrast, classrooms of perceiving type teachers were more informal,

distinguished by movement, noise and socializing among students.

In addition, some research shows connections between the "SJ" and "NF"

combinations of type and the teaching profession. These results lead to interesting

interpretations. Teachers whose type includes the SJ and NF combinations have been

characterized as "stabilizers" and "catalysts", respectively (Clark & Guest, 1995; Keirsey

& Bates, 1985). Stabilizers (SJ) value tradition and are "conservators and maintainers of

the institutions in which they work" (Clark & Guest, 1995, p.20). They additionally note

that this personality profile does not indicate that a "stabilizer" will particularly welcome

institutional change, or will be likely to adopt radically different instructional methods.

Sparks and Lipka note that "Master Teachers", those that are considered by their peers as

exceptional, are also generally found to have a similar respect for traditional ideas

(Sparks & Lipka, 1992). The implications for curricular and instructional change are

obvious in light of this observation from Keirsey and Bates (1985):

The SJ knows as well as others that change is inevitable, necessary, and even, on occasion, desirable; but it should be resisted when it is at the expense of the tried and true, the accepted and approved. Better that change occurs through slow evolution than by abrupt revolution. As conservator of the heritage, the SJ is an enemy of the revolutionary. (p. 44)

In contrast to the tradition-oriented nature of "SJ" teachers, NF or "catalyst"

personalities are interested in making a difference in the world, and tend to strive for

personal and professional growth. Catalysts may devote significant time and effort to

causes, provided that the cause has deep, lasting significance. However, while open to

innovation, NFs may not be likely to initiate change and need time to process and discuss

potential reforms (Clark & Guest, 1995). It should be noted that these types may tend to

13

gravitate toward the humanities and the social sciences, rather than mathematics related

fields (Keirsey & Bates, 1985).

The prevalence of these types among teachers is also interesting. Specifically,

"SJ" and "NF" types are represented in roughly 90% of the teaching force (Hoffman &

Betkoudi, 1981). McCutcheon (1991) reports that the largest subgoup of secondary

teachers in their study were a subset of "SJ" teachers: ISTJ types comprised 14.3%. Their

findings indicate that "SJ" types overall comprised 25% of the secondary teachers

studied, with "NF" at 29% (McCutcheon, 1991). Another study indicates that 42% of

high school teachers are "SJ", and 34% are "NF" (Clark & Guest, 1995). Keirsey and

Bates propose the following: "In any district … over half of the teachers at elementary

and secondary levels pursue an SJ style of life" (1985, p. 46).

Another finding related to the MBTI comes from a study that connects personality

type with teachers' attitudes toward the use of technology (Smith, 1995). This may be

particularly significant, considering that increased technology use has been a feature of

recent mathematics reform recommendations (National Council of Teachers of

Mathematics 1989; National Council of Teachers of Mathematics, 1998). Smith found

that Intuitive/Thinking (NT) types of teachers, those that are creative, analytical, logical,

and imaginative, are more receptive to the use of technology in their classrooms than

"Sensory" types, which are a large proportion of the teaching population. NTs are

characterized as "visionaries", those that are inclined to lead rapid and dynamic change.

Unfortunately, these types make up only 10 to 16 percent of the teaching force, the most

underrepresented of all types (Clark & Guest, 1995).

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So, we should consider that many personality characteristics common among

teachers may potentially conflict with some of the methods of mathematics reform:

cooperative learning, use of manipulatives and technology, investigations, open-ended

problems, whole-class discussions and mathematical discourse. For example, if a greater

degree of noise and disorder may accompany cooperative learning activities and

increased student interaction, one wonders how this might be accepted by many teachers,

who as "judging" types, prefer order and structure (Lawrence, 1982).

Mathematics Teacher Receptivity

Although the research on teacher personalities and tendencies provides useful

background information, it does not speak directly to the beliefs of mathematics teachers

themselves. However, the 1993 Survey of Science and Mathematics Education provides

quantitative data related to the issue of teacher receptivity to reform methods. The study

was based on a national probability sample of science and mathematics teachers and

department heads in the 50 states and the District of Columbia. It provides a wealth of

useful data related to teacher background, textbook usage, and attitudes toward

instructional techniques (Weiss, 1995). Interestingly, the survey was given to teachers

designated as "Presidential Awardees", as well as the national probability sample. These

awardees are teachers who demonstrate the following:

subject matter knowledge and sustained professional growth; an understanding of how students learn mathematics; an ability to generate excitement about mathematics in students, colleagues and

parents; an understanding of the interconnectedness of mathematics and science and the

interconnectedness of all subject matter; an experimental and innovative attitude in their approach to teaching; and professional involvement and leadership (Weiss & Raphael, 1996).

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The data show that Presidential Awardees are much more likely to be in agreement

with the NCTM Standards than their national counterparts (Weiss, 1997). It should be

noted that these awardees were nominated on the basis of these traits, so it should be no

surprise that they strongly agree with the Standards. However, it is interesting to compare

their results with those of the national sample. In examining the two groups' responses, a

number of figures stand out:

Familiarity with the NCTM Standards

Nationally, 56% of 9-12 mathematics teachers considered themselves "well aware

of" the NCTM Curriculum and Evaluation Standards; 40% of 9-12 mathematics

teachers considered themselves "well aware of" the NCTM Professional Standards for

Teaching (Weiss, 1995).

In contrast, 98% of Presidential Awardees teaching grades 7-12 considered

themselves "well aware of" Curriculum and Evaluation Standards; 92% were "well

aware of" the Professional Standards (Weiss & Raphael, 1996).

Nationally, of the teachers who were "well aware of" a particular set of standards,

data were collected regarding their level of familiarity with them:

91% of the national population said they were "well informed about the

Curriculum and Evaluation Standards; 58% considered themselves "prepared to

explain" these standards to colleagues.

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89% said they were "well informed" about the Professional Standards for

Teaching; 55% considered themselves "prepared to explain the Professional

Standards to colleagues (Weiss, 1995)".

From this data, we see that only 33% (or, 58% of the 56% who were "well aware of

the Standards) of national high school mathematics teachers considered themselves

sufficiently knowledgeable about the Curriculum and Evaluation Standards to explain

them to a colleague. Similarly, only 22% could explain the Professional Standards for

Teaching.

However, Stigler and Hiebert (1999) reported a greater percentage of mathematics

teachers being aware of the Standards. Specifically, they found that 95% of teachers

sampled as part of the Third International Mathematics and Science Study (TIMSS) were

"somewhat aware" or "well aware" of the reforms recommended by the NCTM

Standards, and 70% claimed to be implementing these reforms (p. 105). Unfortunately,

after examining videotaped lessons from 81 eighth grade mathematics classrooms, the

authors found that few of these teachers were actually implementing the reforms as

intended. Japanese instruction, in fact, did a better job of emphasizing reform-oriented

ideas such as student thinking and problem solving, multiple solution methods, and

classroom discourse.

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Technology

Nationally, 73% of 9-12 teachers said that "students should be able to use

calculators most of the time" (Weiss, 1995), compared with 93% of the Presidential

Awardees (Weiss & Raphael, 1996).

50% indicated that students working with calculators should "definitely should be

a part of mathematics instruction" (Weiss, 1995), compared with 86% of the

Presidential Awardees (Weiss & Raphael, 1996).

42% of the national population felt well prepared to use computers in instruction;

81% felt well prepared to use calculators (Weiss, 1995).

Views About Mathematics

In contrast to the NCTM Standards, which recommend earlier introduction of

algebraic concepts, Weiss notes that 76% of mathematics teachers indicated that

"students must master arithmetic computation before going on to algebra", whereas only

17% of Presidential Awardees indicated this belief (1996).

Instructional Practices

Nationally, 27% of 9-12 mathematics teachers indicated that students working in

cooperative groups should "definitely should be a part of mathematics instruction"

(Weiss, 1995), compared with 53% of Presidential Awardees (Weiss & Raphael,

1996).

66% of the national population felt well prepared to use cooperative learning

groups.

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62% of 9-12 mathematics teachers said they were well prepared to manage a class

using manipulatives (Weiss, 1995).

Despite these arguably high numbers, Weiss (1997) notes that the survey indicated

that 48% of the time in high school mathematics classes was spent in whole group

lecture/discussion, 14% in small group discussions, and 7% in working with

manipulatives. Furthermore, 94% of high school mathematics classes listened and took

notes during presentation by the teacher at least once a week, and 60% did so on a daily

basis. 98% of high school mathematics classes did mathematics problems from their

textbooks at least once a week, and 86% did so on a daily basis.

In some ways, this research supports the implications of the personality type data.

We have more confirmation that mathematics teachers, as a whole, have not acted on

many of the reform recommendations regarding instructional practices, as was also found

by Stigler & Hiebert (1999). Even when these ideas are accepted, implementation is still

problematic, as case studies discussed below will illustrate. This is not to suggest that

mathematics teachers are incapable of incorporating cooperative learning, whole-class

discussions and mathematical discourse; nor are they universally opposed to these ideas.

However, we see that there is still, at times, a disconnect between these ideas and

teachers' receptivity to using them.

Providing support to the 1993 quantitative data from Weiss, numerous case

studies have been conducted documenting teachers' attempts to implement the NCTM

Standards, as well as their attitudes toward reform-oriented methods (Benbow, 1993;

Chauvot & Turner, 1995; Clarke, 1997; Cooney & Wilson, 1995; Eggleton, 1995;

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Frykholm, 1996). These qualitative studies provides indications of the conditions under

which teachers may be receptive to reform-oriented curricula and instructional practices.

Three themes emerged from an examination of these studies: beliefs about the nature of

mathematics, beliefs about the teacher's role, and reflective practices.

Beliefs About Mathematics

One theme coming from these studies concerns the mathematical beliefs of the

teacher prior to teaching. We see a difference between teachers that consider mathematics

a question of absolute right or wrong, versus those who view mathematics a dynamic

field, influenced by context and personal factors. Specifically, the former group, those

with an “absolutist” view of mathematics, have a more difficult time accepting the

principles of the NCTM Standards than does the latter group, who have a “relativistic”

view (Cooney & Wilson, 1995).

For example, Eggleton reports a pre-service teacher that considered mathematics

a "set of facts", or "number crunching," and himself as the center of mathematical

authority for his students (1995, p. 2). Similarly, Cooney (1995) documents a preservice

teacher that held an "assembly line" view of mathematics, with a rigid filtering system of

what she accepted as evidence of learning and good practice (p. 3). Predictably, these

teachers had difficulties realizing the vision of teaching espoused in the NCTM

Standards, as their initial views about mathematics seem contrary to the spirit of those

documents (National Council of Teachers of Mathematics, 1989; National Council of

Teachers of Mathematics, 1991; National Council of Teachers of Mathematics, 1998).

20

Again, this is not to suggest that teachers with an absolutist view of mathematics

are immovable in their beliefs or are incapable of realizing a reform-oriented vision of

teaching. Furthermore, it would be a mistake to assume that one’s beliefs are strictly

absolutist or relativistic. Yet, we see in these studies and others that when teachers hold a

rigid, narrow view of mathematics, that nontraditional teaching develops far more slowly

(Benbow, 1993; Clarke, 1997). Unfortunately, we should also note that this view is not

uncommon in preservice mathematics teachers (Eggleton, 1995).

By contrast, teachers that hold a primarily relativistic view of mathematics have

an easier time implementing the Standards. Another preservice teacher studied by

Cooney had such a view of mathematics, believing in emphasizing the connectedness of

mathematics to the real world for his students. In addition, he saw mathematics from a

broad perspective, as opposed to narrowly defining the field as the sum of his

postsecondary mathematics course topics (1995). Clarke (1997) describes a similar

situation with a veteran teacher. In these cases, the teachers' beliefs proved flexible,

which allowed them to adapt their teaching to learners' needs in the context of

nontraditional methods.

Beliefs About the Teacher's Role

A second notable theme concerned teachers' beliefs about their role in the

classroom, and the ability to critically examine it. For example, Chauvot and Turner

(1995) studied one preservice teacher who saw her role as providing a "non-intimidating,

non frustrating, interesting and motivating" environment for her students. As she

proceeded through her student teaching program, she was encouraged to implement

21

reform-oriented methods, which in turn strengthened her beliefs. These beliefs led her to

increasingly use multiple teaching methods and group work and emphasize problem

solving. In addition, this teacher began to negotiate her role in the classroom, specifically

with how much direction she should provide the students.

"Telling" is a theme that appears in other studies as well (Benbow, 1993; Clarke,

1997). Chazan and Ball argue that one aspect of the teacher's role is to use discussion-

intensive teaching to support an atmosphere of "intellectual ferment" (Chazan & Ball,

1995, p. 18). In this environment, "In small and large groups, students are to present their

ideas and solutions, explain their reasoning and question one another" (p. 1). As a result,

teachers must reframe their role as the center of mathematical authority that has

traditionally supported telling the students whether they are correct or incorrect.

Those that were most successful in implementing reform ideas into their

classrooms were those that were able to "step back", in the words of one teacher. Letting

students struggle with problems and resisting the urge to squelch mathematical discussion

by giving the answer became a "liberation" for this teacher. For him, the teacher's role

became a question of "to tell or not to tell" (Clarke, 1997, p. 288).

Of course, Chazan and Ball (1995) argue that mathematical discourse involves

more than "telling or not telling," but it must be noted that the willingness to explore this

aspect of teaching is an important indicator of success with Standards-based materials. In

contrast, we see that many teachers still hold metaphors for teaching that imply an

authority or controlling relationship (Fleener, 1995). This is consistent with personality

type data regarding teachers, but could be problematic for the widespread use of reform-

oriented methods.

22

For example, Gregg documented the difficulties of one mathematics teacher in a

school serving several counties with enrollment of about 1200, located a Midwestern

town of about 5000 residents (Gregg, 1995). The emphasis the teacher placed on

maintaining control in her general mathematics class led to several unfortunate results:

The teacher's expectation that the students would get out of control easily

influenced her behavior, causing her self-described "meanness." The students reacted

to this by being more disruptive. This behavior further influenced her view of the

students, causing her to be more "mean". Thus, a self-perpetuating cycle was

initiated.

When the teacher would have trouble keeping the class under control, she would

stop asking leading questions that she used in other classes to engage students.

Although she was trying to limit the opportunities for breakdowns in control, this

contributed to the students' boredom, which aggravated discipline problems. This

started another contradictory cycle.

Contradictions came into play when the teacher attempted to implement a general

math program written in the spirit of with the NCTM Standards emphasizing problem

solving. The program was discontinued because the "kids couldn't handle the

freedom" (p. 590).

The Ability to Reflect

In virtually every one of the case studies examined, the authors cite the ability to

reflect on teaching practices as a crucial component of success (Benbow, 1993; Chauvot

23

& Turner, 1995; Clarke, 1997; Cooney & Wilson, 1995; Eggleton, 1995). Eggleton

writes, "the difference between instruction and indoctrination was the provision of

opportunities for the individual to radically examine his/her belief systems." (1995, p. 4).

These cases illustrate the truth in these words. Further, Fullan and Stielgelbauer (1991)

write,

Beliefs guide and are informed by teaching strategies and activities; the effective use of materials depends on their articulation with beliefs and teaching approaches and so on. Many innovations entail changes in some aspects of educational beliefs, teaching behavior, use of materials and more. Whether or not people develop meaning in relation to all three aspects is fundamentally the problem. (p. 41)

Case study research suggests that those who were able to reconcile their beliefs

about mathematics and the role of the teacher with reform recommendations were

consistently able to reflect on their practice, and reconstruct it accordingly.

Unfortunately, case studies also suggest that those who had neither the inclination nor the

opportunity to engage in this sort of reflection struggled with their teaching experience.

Departmental Receptivity

Although an examination of the characteristics of individual teachers may be

useful, this knowledge may be of limited importance. Waugh and Punch (1987) suggest

that the change process is additionally dependent on factors such as: organizational

features, the politics surrounding the change, and the type of change being implemented

(p. 241). Furthermore, Baldridge and Deal (1975) found that organizations often have

deep roots in history, and suggest that teachers are not likely to be strongly receptive to

any proposed or attempted implementation of a change that is in direct conflict with the

24

traditional values of a school or school system (p. 15-16). Therefore, when considering

the implementation of a new program at the department level, an examination of the

characteristics of similar departments is appropriate.

Specifically, it may be argued that mathematics reform documents are a response

to a strikingly durable tradition that has evolved in many American mathematics

departments. This is the traditional school approach to mathematics, such as the routine

of checking homework, teacher presentation of material, students working examples, and

assignment of seatwork (Cobb, 1992; Stigler & Hiebert, 1999). One wonders what factors

affect the overall environment of a mathematics department, indicating whether it can

break the stranglehold of the "school mathematics tradition".

Information about mathematics departments and the NCTM Standards is

appearing in the form of both quantitative and qualitative data. Garet and Mills report

results of a 1991 survey given to mathematics department chairs in all public,

comprehensive high schools within 100 miles of Chicago (Garet & Mills, 1995). The

survey asked respondents to provide information about practices in first-year algebra

classes, and several findings provide interesting background for this question.

For example, average consistency with practices recommended in the NCTM

Standards documents among schools is increasing, but the variation between schools is

dramatic. Furthermore, this variation is increasing over time. In part, this is associated

with demographic characteristics in the schools. Generally, suburban schools reported the

level of practice most consistent with the Standards in 1991, but by 1996, urban centers

predicted greater alignment of their instructional methods.

25

In addition, the study documented continued prevalence of ability grouping, a

practice often criticized as inequitable (D'Ambrosio et al, 1992; McKnight et al, 1987;

National Council of Teachers of Mathematics, 1989). These numbers indicate that ability

grouping was associated with school size: 70% of small schools with enrollment under

500 reported just one level of Algebra 1. In contrast, 10% of large schools with

enrollment over 2000 report only one level of algebra. This is probably no surprise, as

smaller schools have fewer resources to expand curricular offerings. Yet, it is interesting

to note that as schools increase in population, the number of course offerings increase.

Curriculum thus becomes less compact, resembling a wider smorgasbord of choices for

students. However, Gutierrez (1996) notes the importance of a compact, less

differentiated mathematics curriculum in encouraging high levels of success in a wider

range of students.

Although these quantitative data are illustrative, specific case studies provide

more detail and allow one to examine the real character of successful mathematics

departments. For example, Meador documents the successes and failures of a

mathematics department that has been engaged in reform efforts since 1977 (Meador,

1995). In addition, Gutierrez studied the beliefs and practices of several mathematics

departments that succeed in encouraging more students to take mathematics at higher

levels by the end of their senior year (Gutierrez, 1996).

Although these studies did not specifically address implementation of reform-

oriented curriculum as defined for this paper, characteristics of these programs are

remarkably consistent with such recommendations. Therefore, it is perhaps no surprise

that these departments have implemented many of the key components of reform

26

documents. In particular, some of these are an innovative curriculum, nontraditional

teaching methods, and the use of appropriate technology (Mathematical Sciences

Education Board, 1989; Mathematical Sciences Education Board, 1990; National Council

of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991;

National Council of Teachers of Mathematics, 1999). However, the question is, what

aspects of the department allow the implementation and sustained success of these

components?

Professional Activity of Teachers

According to these studies, one common element in successful mathematics

programs is the teachers' involvement in professional activities. These departments allow

and encourage teachers to interact with each other, and with teachers in other schools.

Through workshops, involvement in professional organizations, or informal extraschool

communities, teachers are exposed to new ideas and are able to share information.

Meador writes, "Teachers enjoyed collaboration and interaction with other teachers in

and out of their department, and the network of support they developed outside of the

school was crucial to their continued emphasis on trying new ideas in the classroom"

(1995, p. 13).

Leadership

Related to this, Garet and Mills (1995) note the crucial role played by the

department chair in guiding and supporting change. Specifically, the Department Chair's

involvement in professional organizations is significant. 75% of chairs in the 1991 survey

27

sample belong to the NCTM, and 25% attended that organization’s annual meeting in

1991. However, professional organizational involvement was differentiated by

geography, similar to the variation in practices associated with the NCTM Standards.

Roughly one-half of the chairs in Chicago, Milwaukee, and their suburbs attended the

1991 meeting, but only 14 percent of chairs in smaller cites and 3 percent of the chairs in

rural communities attended (Garet & Mills, 1995).

Similarly, a majority of the chairs in the sample report being quite familiar with

the Standards and strongly agree with the reforms proposed. Yet, while 75% of the chairs

in the suburbs reported being quite familiar with this document, only one third of the

chairs in rural districts reported this. This research suggests that department chairs that

are active in the professional community are more aware of curriculum and evaluation

reforms, and are more likely to report departmental teaching practices espoused by the

Standards documents (Garet & Mills, 1995).

Mathematics for All

Although not easily quantifiable, successful mathematics departments also share

another characteristic supported by reform documents: active commitment to all students

(Mathematical Sciences Education Board, 1989). In these departments, an atmosphere of

"Mathematics for All" is a reality, not an empty slogan. Gutierrez writes, "Despite their

students' weaknesses teachers in (these) departments tended to concentrate on students'

strengths to aid in the teaching and learning process" (1996, p. 511). Furthermore,

teachers held high expectations that all students could handle a rigorous curriculum.

28

In contrast, departments that foster non-constructive, negative conceptions of their

students may be unsuccessful implementing Standards-based curriculum. Examples of

these attitudes may be, “too many students aren’t ready to learn the material we have to

teach them”, or, “students are unwilling to learn” (Gutierrez, 1996, p. 511). This

represents a "clash of cultures" line of reasoning commonly used to explain students' lack

of achievement: the irresistible force of the students' home culture meets the immovable

object of the traditional school culture (Gregg, 1995, p. 598).

Control and Authority

It may also be important to critically examine the school culture and the status

afforded teachers, based on their ability to maintain authority and discipline. In the

successful departments, teaching success is not based on authority relationships, but on

student achievement (Gutierrez, 1996; Meador, 1995). In a study by Gregg, an emphasis

on discipline and control was a main concern with the both teacher and the principal

(1995). Again, the only serious reform effort undertaken was abandoned in less than a

year, as a result of perceived discipline problems.

This is consistent with conclusions drawn from a study that examined the factors

that influenced teacher induction (Schempp, Sparkes, & Templin, 1993). This study

found that teachers were accepted on the basis of their classroom management skills, and

that teachers' control over students is a taken-for-granted assumption in schools. In fact,

one of the beginning teachers studied by Schempp et al. won his mathematics position

because of his classroom management abilities, rather than his meager subject matter

knowledge. He writes, "Success or failure, bluntly put, resided in the teachers' control

29

over students: the greater the level of control, the greater the level of success" (Schempp

et al., 1993, p. 459).

Change Embedded in the Culture

Finally, mathematics departments that are successful at implementing

nontraditional curriculum are described as having change embedded in the culture, as

demonstrated by a department that began implementing reform-oriented curriculum and

instructional practices over ten years prior to the publication of many of the national

mathematics reform documents. Meador writes, "These teachers began their struggle

without the benefit of school reform literature… The motivation to change was theirs and

theirs alone" (1995, p. 1). In this department, change was talked about, looked for and

expected.

Similarly, Gutierrez describes a "commitment to a collective enterprise" common

to departments who encourage high levels of students mathematics success (1996, p.

507). In addition to an emphasis on professional development, teachers in these

departments communicated actively and practiced collective decision-making.

Final Thoughts

After examining mathematics teaching in the United States, Japan and Germany,

Stigler and Hiebert (1999) note a "distinctly American way of teaching" (p. 11), and

suggest that we are often unaware of some of the most widespread attributes of teaching

in our own Western culture. They argue that the cultural nature of teaching may explain

30

why it has so far been so resistant to change, but recognizing this nature may provide

insights as to what must be done to improve education.

Although Stigler and Hiebert (1999) found little variability among American

teaching methods, attempting to change this teaching culture will likely be less

predictable. Conditions under which reform is welcomed, rejected, and when it stalls will

be specific to the local situation shaped by the students, the community, the teachers, and

school administration. Because educational change is such an incredibly dynamic and

difficult process to model, the recurring theme of "culture" could be a crucial organizing

principle. Fullan and Stiegelbauer (1991) write,

Thus the meaning of change for the future does not simply involve implementing single innovations effectively. It means a radical change in the culture of schools and the conception of teaching as a profession…Cultural change requires strong, persistent efforts because much of current practice is embedded in structures and routines and internalized in individuals. Yet cultural change is the agenda. (p. 142-143)

Therefore, if departments do not reflect on and critically consider institutionalized

practices such as tracking, emphasizing grades, and teaching in ways that reinforce the

school mathematics tradition, reform efforts will likely fail. They may never truly be

attempted in more than superficial ways. Since I read of success stories in the research, I

realized there was a possibility that the Adderley mathematics department would add to

these successes.

31

CHAPTER THREE

METHODOLOGY

For this study, my goal has been to collect data related to the culture of the

Adderley High Schools mathematics department, and specifically the receptivity of the

teachers to this curricular and instructional change. To this end, I have noted some of the

departmental history, traditions, and beliefs that have shaped this reform effort. My hope

is that this may help illustrate the conditions under which teachers and departments in

general are ready to accept such a shift in direction.

These data were collected in two stages. The first stage was conducted in

September 1999, and the second in February 2000. A written survey accompanied the

first stage of interviews (see Appendix A). In both stages combined, 15 teachers out of

the 22 in the department were interviewed; six of these teachers were interviewed at both

stages. Because CPMP is not yet fully phased in at all levels, some teachers have not yet

taught these materials, but these interviews were conducted almost solely with Core-Plus

teachers.

The Program

Before discussing the specific nature of these interviews, a description of

Adderley’s program is necessary. Certainly, the foundation is the Core-Plus Mathematics

Program, although the department has modified CPMP considerably by connecting it to

the "Accelerated Mathematics Program". Approximately two days per week, each teacher

suspends CPMP instruction and the class interacts solely with Accelerated Mathematics,

which is a computer-based drill-and-practice instructional program; more detailed

32

descriptions of both of these programs are given below. Another variable to consider

relates to a district-wide technology initiative: the model technology classrooms. This

affects only two mathematics teachers per campus, but the impact could still be

significant, and is further evidence of the many complex changes taking place in and

around the department.

The Core-Plus Mathematics Project

Developed with funding from the National Science Foundation, the CPMP is a

high school mathematics curriculum designed around the principle of "mathematics as

sense making". From the preface to the textbook series, the authors discuss this concept:

"Through investigations of real-life contexts, students develop a rich understanding of

important mathematics that makes sense to them and which, in turn, enables them to

make sense out of new situations and problems" (Coxford et al., 1998).

Guided by this principle, the program has several key features that set it apart

from a "traditional" high school mathematics curriculum. For example, each year does

not focus on one particular mathematical topic, such as algebra or geometry. Instead, four

content strands are woven together each year of the program: Algebra and Functions,

Geometry and Trigonometry, Statistics and Probability, and Discrete Mathematics.

Furthermore, the program emphasizes mathematical modeling through data

collection, interpretation, prediction, and simulation, and central topics are designed to be

accessible to all students. The use of graphing calculator technology is standard at all

levels, and the curriculum developers give a number of reasons for this. First, the

graphing calculator supports emphasis on numeric, graphic, and symbolic

33

representations, a theme repeated throughout the course. In addition, the calculator allows

students to focus on mathematical reasoning rather than computational proficiency in

situations where this is appropriate (Schoen, Hirsch, & Ziebarth, 1998).

Finally, the designers of the curriculum stress that students should be actively

engaged in the material. That is, a high priority is placed on the use of collaborative

activity, as opposed to teacher exposition. The students are to "explore, conjecture,

verify, apply, evaluate and communicate mathematical ideas" (Coxford et al., 1998, p. x).

Therefore, reading, writing, and active mathematical reasoning and modeling on the part

of the students are crucially important to instruction and assessment.

The authors provide a rich description of the intended instructional pattern:

Most classroom activities are designed to be completed by students working together collaboratively in heterogeneous groupings of two to four students…The launch phase promotes class discussion of a situation and of related questions to think about, setting the context for the student work to follow. In the second or explore phase, students investigate more focused problems and questions related to the launch situation. This investigative work is followed by a class discussion in which students summarize the mathematical ideas developed in their groups, providing an opportunity to construct a shared understanding of important concepts, methods and approaches. Finally, students are given a task to complete on their own, assessing their initial understanding of the concepts and methods (Coxford et al., 1998, p. xi-xii).

See Appendix B for specific textbook examples.

Research conducted on the mathematical achievement of Core-Plus students, as

well as on the affective outcomes of these students, indicates that the curriculum can be

successfully implemented in these areas. Specifically, one study found that CPMP

students outperformed traditionally taught students on a number of standardized tests

(Schoen et al., 1998). Other research has suggested that CPMP students are more positive

34

about certain aspects of the curriculum and of their classroom experience than students in

traditional classes in the same schools (Schoen & Pritchett, 1998).

The Accelerated Mathematics Program

In contrast to the CPMP emphasis on exploration and communication, the

Accelerated Mathematics Program is decidedly more skills-based. "Ack Math," as it is

known in the district, is essentially a computer program designed to reinforce basic

mathematical skills. As this is perceived as a weakness of the CPMP curriculum within

the department, the department chair has strongly suggested that teachers split

instructional time between CPMP and Ack Math. As noted above, Mr. Blakey

recommends that students engage with the Accelerated Mathematics Program two full

days per week.

The software has a database of hundreds of skills objectives, most of which are

associated with traditional Pre-Algebra, Algebra I, Geometry and College Algebra course

content. It presents the students with an endless supply of multiple choice problems that

provide practice in each objective. However, the department has made an attempt to

select objectives aligned with the Core-Plus sequence. For example, as students are

engaged in the "Exploring Data" lesson in CPMP, during Ack Math time they might

encounter an objective that reinforces finding the median from a stem and leaf plot. See

Appendix C for specific examples of Accelerated Mathematics objectives and problems.

Unique features of this program are that students work at an individual pace, and

receive immediate feedback on their work. Each classroom is equipped with a laptop

computer that is connected to a scanner that grades the students' worksheets as they

35

complete them. Students are able to practice and receive feedback as often as they like

before they are tested on several objectives at once. Once a student tests at an 80% level,

the computer considers these objectives met, and provides the student practice with a new

set of problems related to new objectives. Since students complete the objectives

individually, a class of 25 students might realistically be engaged in meeting 25 separate

sets of objectives. Therefore, the teacher's role is often to circulate around the class and

provide individual guidance when needed.

Model Technology Classrooms

The Model Classroom Project is a district-wide technology initiative that has

created a number of specialized technology classrooms in both Adderley High School

buildings. The mathematics department currently has one technology classroom per

campus, and two teachers share each classroom. Each room contains seven student

computer workstations, a computer screen projection unit, printer, scanner, and flex-cam

for broadcasting live images on the Internet. In addition, each classroom is wired for

rapid T-1 access to the internet and computers are equipped with standard word

processing and spreadsheet software, presentation tools, web design software, and

internet browsers. The mathematics department has several content-specific applications

such as Geometer's Sketchpad. The 50 teachers in the district who share the 20

classrooms participate in ongoing training, and each is loaned a district-owned laptop

computer for his/her personal and professional use.

The design of the Model Classroom Project was chosen by the district for several

reasons. First, it is viewed as a cost-effective way to fully integrate technology into the

36

learning process. Furthermore, the district believes that the design allows for students to

be actively engaged in their own learning, working in groups or individually to carry out

assignments in research, design, problem solving, and presentation. The mathematics

teachers who share these rooms have periodically enhanced the CPMP/Ack Math

program with the use of spreadsheets and Internet access.

Personal Reflections

The actual introduction of the CPMP materials to the department has a simple

origin. Initially, these materials were discovered and introduced to Adderley by the

current department chair, Mr. Blakey. Before moving into his administrative position,

Blakey was a teacher in the department, and learned about the CPMP curriculum at a

conference. He began to experiment and had some success with elements of the program

in his own Algebra S class. When he became Department Chair, it was a natural step to

begin the pilot program.

However, as encouraged by CPMP as I was, I anticipated problems if the

curriculum were implemented on a department-wide scale. My concerns echoed several

of the criteria teachers use in assessing any given change, epitomized by the question

“Why should I put my efforts into this particular change?” (Fullan and Stiegelbauer,

1991, p. 127). According to Fullan and Stiegelbauer, teachers question how any change

will affect them personally “in terms of time, energy, new skill, sense of excitement and

competence, and interference with existing priorities" (p. 128). For teachers accustomed

to a traditional curriculum, this new curriculum might be extremely demanding in all of

these areas.

37

For example, upon inspection of the course materials, it is immediately apparent

that the CPMP text does not look like a traditional text (see Appendix B). One does not

find a list of exercises and lists of problems to be completed by students during "seat

work". Rather, the textbook is written more in paragraph form, divided into

"Investigations" that emphasize student experimentation and exploration. In fact, the use

of cooperative learning methods is often specified directly in the context of a problem.

While some teachers at Adderley were familiar with these strategies, others were not. In

addition, I knew that the heavy emphasis on the use of the graphing calculator would be

problematic for some teachers, and that extensive training would be necessary.

Furthermore, the program covers different mathematical topics than those covered

in a traditional sequence. For example, the first course begins with a unit in statistics. The

students collect data, construct histograms and box plots, and discuss variation. Later, the

concept of slope is developed in the context of linear regression. Many of the teachers in

the pilot program agreed that they had first encountered these statistical topics in college.

So, this material demands that teachers may initially have to learn along with the

students, which I took as time-consuming, and thus problematic.

In addition, the fact that mathematical topics are organized differently in CPMP

than in a traditional course sequence could have affected teachers' willingness to

implement the curriculum. Again, the text is organized in "strands" that are repeated from

year to year. For example, the "Algebra and Functions" strand extends across several

years of instruction. Therefore, topics traditionally associated with a first year algebra

course may not be developed fully in the first year of CPMP. I predicted that this would

be disconcerting for many teachers.

38

A specific incident illustrates the discomfort some felt with this aspect of the

program. Before implementing the pilot course, the department chair organized a summer

training session with the initial six pilot teachers. It was led by the department chair of a

school in Michigan that had extensive experience with the CPMP materials, having field-

tested the program for the curriculum publisher. During the session, an Adderley teacher

casually noted that he could not find where the "order of operations" was covered in the

first year text. The Michigan teacher thought and replied, "I don't think we taught that last

year. It didn't come up." This is consistent with recommendations that a strict adherence

to teaching a rigid sequence of topics may not be beneficial for all students (National

Council of Teachers of Mathematics, 1998).

Yet, I think that the statement made by the Michigan instructor came as a shock,

and caused quite a lively discussion. The thought that such a common topic wasn't taught

because it "didn't come up" sparked nervous laughter, and was emblematic of the change

that would be demanded of many. It was an uprooting of the familiar sequence of topics

that we had become comfortable with as teachers, and it was intimidating to some.

Another tension perceived at that time by some in the department related to

teachers’ years of experience. Specifically, the teachers were split almost evenly between

those with less than 10 years of professional experience and those with 20 years or more.

Certainly, those that seemed to be open to innovation were not only novice teachers, and

those that tended toward traditional methods were not exclusively the veterans. However,

departmental issues and discussions about instruction and assessment often fell along

these lines. Chapter 4 discusses the implications of this characteristic of the department

more fully.

39

Finally, I was concerned by the assertion that teachers question how changes

affect them in terms of existing priorities (Fullan & Stiegelbauer, 1991). Specifically, I

perceived a philosophy among some teachers in the district that very clearly distinguished

"Honors Students" from "Non-Honors Students". Even if a student displayed a healthy

work ethic and the willingness to be challenged, this philosophy often made moving into

the honors lane more difficult than moving into other lanes. Several of my colleagues and

I joked that a touch of elitism characterized the department at that time; allowing "Non-

Honors" students into "Honors" classes might contaminate the purity of these classes.

Whether this was accurate or not, many influential teachers in the department seemed

satisfied with this tracked, traditional course structure.

This is not to say that the department as a whole was closed to innovation or

experimentation. The Geometry/Chemistry Honors Plus program is an example of this. In

this program, Geometry and Chemistry are team-taught over three consecutive periods,

allowing the teachers to introduce long term projects incorporating Internet website

design. Furthermore, the department had experimented with alternative curricula, such as

the University of Chicago School Mathematics Project (UCSMP), on a larger scale

before 1996-1997. The UCSMP implementation had not survived, although the

department chair during my first year at Adderley strongly emphasized technology, the

use of cooperative learning strategies, and authentic assessment materials. Again, some

teachers in the department agreed that these strategies were beneficial, and employed

them extensively.

Changes are problematic for any mathematics department accustomed to

traditional teaching. My concern primarily stemmed from the perception of mathematical

40

elitism in the department. Again, this curriculum is consistent with a vision of

"mathematics-for-all," rather than "mathematics-for-the-elite” (D’Ambrosio et al, 1992;

Mathematical Sciences Education Board, 1989). I asked myself: Could this really be

accepted as "school mathematics"? Could this program possibly replace a "pure

mathematics program"? Would the teachers in this department be willing to give all

students the opportunity to engage in significant mathematics regardless of their

computational skill? However, despite these questions, I knew that these teachers were

first and foremost professionals, ultimately wanting the best for the students.

Related to the priority placed on doing what's best for the kids, I return to Fullan

and Stiegelbauer (1991). Another criteria teachers use in assessing any change is, “Does

the change potentially address a need? Will students be interested? Will they learn? Is

there evidence that the change works?” (p. 127). I believe that, although these teachers

may have been unsure of how a curriculum change would have affected them personally,

they certainly sensed a need to be addressed. It is possible that concerns about the

traditional curriculum may have been forming within the department for some time. In

the minds of a number of teachers, disturbing trends were beginning to develop, and the

old curriculum was beginning to be cast in doubt. One teacher later noted, "(the

traditional curriculum) just wasn't working for 90% of the students."

Although neither quantitative nor qualitative data were collected at the time, there

was anecdotal evidence that the exisiting curriculum was not leading to the positive

outcomes hoped for by the school. Through conversations with those in the department, I

noted some of these perceptions. According to these teachers, the traditional curriculum

wasn't working because over the last few years:

41

Fewer students were taking more than the required number of mathematics courses.

Fewer students at all levels seemed motivated to complete any class work. An increasing number of students at all levels seemed content to do the minimum

required work to pass. Absences were more common. An increasing number of students failed mathematics classes.

It is important to note that these perceptions may or may not be grounded in fact. It

is not clear, for example, that students were actually less motivated than in years past; no

attitudinal data were collected. Furthermore, if student motivation had actually decreased,

it is still not clear whether was due to the curriculum. Could these same students have

been motivated with the same traditional materials, but with different instructional

strategies? Again, this is not clear, but the fact that these perceptions were common in the

department is significant. It seems that these perceptions did, in fact, influence and

support the decision to move in a different direction.

In particular, one teacher expressed a common perception related to the

ineffectiveness of the traditional curriculum. When this teacher started at East Adderley,

she estimated the campus served some 2000 students. Enrollment later dropped to

roughly 1400, and she estimated it correctly at around 1800 students in 1999. When she

started, the students were primarily of Italian, German, and Polish descent. Now the

student body is 33% Latino, and since the break up of the Soviet Union, the school serves

a larger population of students from Eastern Europe. As a result, she said, "there have

been significant changes in what people bring to us when they come here". That is, some

students have been exposed to little mathematics and little technology. In addition, some

students have a very strong mathematics background but still have had no experience

with technology. Furthermore, far more students speak English as their second language.

42

Many teachers at Adderley see the CPMP/Ack Math program as an attempt to

address needs of this changing student body, an "attempt to level the playing field".

Because the student population at Adderley High Schools is becoming more culturally,

ethnically, and racially heterogeneous, a traditional mathematics curriculum did not

address the needs of these students. The new curriculum is perhaps an attempt to give all

students an opportunity to succeed, in spite of variation among students in English

speaking ability, and knowledge of mathematics and technology. Seen in this light, these

teachers had a very strong impetus to make the new program work. It would "address a

need."

The purpose of the preceding discussion has been to illustrate the background

knowledge I had prior to conducting this research. I had knowledge about the faculty, the

students, the administration, the departmental culture, and about the new program itself.

Equipped with this information, I returned to the department twice to collect data for this

study. The first stage of data collection occurred at the beginning of the 1999-2000

school year, and the second occurred shortly before the end of the first semester.

Data Collection: Stage One

The first stage of data collection was conducted at the beginning of the school

year because I sought to document teachers' perceptions before they had a chance to

become accustomed to their schedule, students, and, most importantly, the new program.

Many of those I interviewed had never taught CPMP, and I wanted to note their

preliminary impressions and expectations. I administered written surveys to 12 teachers

43

of the Core-Plus Curriculum, and I conducted audio taped interviews with 9 of those

teachers.

For this first stage, I focused primarily on teachers' perceptions of one aspect of

the classroom environment. Since a comprehensive discussion and analysis of the many

levels of paradigm shifts that might occur in the classroom would be a monumental task,

I specifically centered the interview questions on their beliefs about authority relations

and control in the classroom.

By “control”, I mean the tendency of many mathematics teachers to micromanage

and act as dictators (however benevolent) of nearly every aspect of the classroom

environment. Traditionally, power relations in the mathematics classroom have been

entirely top-down, where authority is exerted from on high, and all knowledge emanates

from the teacher. All too often, the teachers' desk sits like an altar at the front of the

room, and the students, congregation-like, sit in orderly rows and columns.

Alluding to the control issue, the Principles and Standards for School

Mathematics Discussion Draft states, “Knowing the capacity of the entire class to move

forward from a single student’s idea and juggling the tradeoffs in allowing an

unanticipated digression are part of orchestrating good mathematical discourse” (National

Council of Teachers of Mathematics, 1998, p. 33). Notable here is the suggestion that

students have influence in the direction of the class, that students be empowered in part to

guide their instruction. With this in mind, “classroom management” and “on-task

behavior” take on entirely different meanings. So, it would seem that teaching in the

spirit of the NCTM Standards, for example, implies that the teacher must relinquish some

control and provide the students the opportunity to grasp it.

44

However, the noise and surface disorder of a “student-centered” classroom have

long been the hallmarks of what a good teacher is supposed to avoid. Sizer calls this

unspoken pact between students and teacher to present a facade of sterile orderliness a

“Conspiracy for the Least”, the least hassle for anyone (Sizer, 1992, p. 156). That is, the

teacher demands little effort from the students, and in turn, the students cause the teacher

few discipline problems. Unfortunately, research as shown that these bargains are

common (Cusick, 1983; Powell, Farrar, & Cohen, 1985), and protect the status quo by

"using power to form a negative pact that reduces pressure for reform" (Fullan &

Stiegelbauer, 1991, p. 180).

Furthermore, Romberg notes, “There is an inexorably logical sequence when the

acknowledged work of teachers is to transmit the record of knowledge; the most cost-

effective way to accomplish this is through exposition to a captive audience...And that

exposition cannot happen unless there is control" (Romberg, 1994, p. 7-15).

Because the CPMP is designed to be more student-centered than a traditional

curriculum, I hypothesized that teachers would have to be prepared to relinquish a

measure of authority for a shift toward a reform-oriented vision to be fully realized. In

fact, I hypothesized that the release of a measure of control by the mathematics teacher,

permitting other voices in the classroom to be heard in a meaningful way, is a particularly

crucial element in such a shift.

As a result, I thought it would be fruitful to investigate the relationship between

mathematics teachers’ concepts of mathematics as a static body of knowledge to be

transmitted and classroom power relations. If mathematics itself is considered by a

teacher to be the ultimate non-negotiable authority, a dictatorial relationship between

45

teacher and students may be a natural outward expression of this scheme. Perhaps this

leads to a self-sustaining mathematical culture in which the authoritarian tradition of

mathematics teaching is handed down and recycled generation after generation.

Therefore, these perceptions would directly affect the implementation of the

Core-Plus curriculum. To explore this area of teachers' beliefs, the survey and the

interviews were designed to gather data regarding teachers' intended instructional

methods, beliefs about mathematics teaching and learning, and perceptions of the

teacher's authority in the classroom.

Data Collection: Stage Two

The second stage of data collection was scheduled to occur immediately

following the first semester, and capture whether or not the teachers' attitudes had

changed after actually teaching the materials. Specifically, I intended to interview the

same teachers a second time, documenting their perceptions of the Core Plus classes, and

whether they were able to sustain an environment that facilitated reform-oriented

teaching and learning.

However, the results of the first round of interviews were particularly interesting,

in that I detected attitudes toward authority and control that I never expected. This will be

developed further in Chapter 4, but I suspected that a very intriguing situation was

developing in the Adderley High School mathematics department, and I sensed that the

department's culture was undergoing a palpable change.

Exploring the cultural aspects of the department, Weissglass (1992) defines

culture as follows:

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Culture is the attitudes, beliefs, values and practices shared by a community of people which they often do not question, are often unstated and which they may not be aware of. (p. 196)

Furthermore, Deal and Peterson (1999) offer this definition of school

culture:

School cultures are complex webs of traditions and rituals that have been built up over time as teachers, students, parents, and administrators work together and deal with crises and accomplishments. (p.4)

Using these definitions, one can see the potential for cultural change in the

department. Weissglass notes the importance of unstated attitudes and beliefs in a

community, and Deal and Peterson suggest that crises and accomplishments can greatly

effect these aspects of culture. Certainly, the Adderley mathematics department faced

obvious crises in the form of a challenging new curriculum, which required new

mathematical knowledge, pedagogical skills, and proficiency in technology. In light of

Fullan and Stiegelbauer’s assertion that “cultural change is the agenda” (1991, p.142-

143), the effect of the new curriculum on the “complex webs of traditions and rituals” in

the department became a more pressing concern than authority and control issues.

Attitudes, overall feelings, and general behavior intentions have been shown to

account for a large percentage of teachers’ receptivity to change, and a causal link has

been established between these three components. That is, overall feelings influence

attitudes, and both feelings and attitudes affect behavior intentions (Waugh & Punch,

1987). Thus, the unstated attitudes, beliefs, values and practices uncovered in the first set

of data led me to focus the second stage on developing a picture of the overall culture of

the mathematics department, as perceived by the teachers.

47

To this end, the work of Gutierrez (1996) had considerable influence on the

second stage of data collection. Her research examined "the beliefs, practices, and general

teaching cultures of mathematics departments that are successful in getting students to

take more mathematics, and higher levels of mathematics, by the end of grade 12" (p.

495). Of course, these were many of the same goals of the Adderley mathematics

department in its reform efforts, and I sought to make connections between this

department and the departments Gutierrez studied.

Specifically, Gutierrez describes a framework she calls "Organized For

Advancement" (OFA). An OFA department is indeed successful in encouraging more

students to take and succeed in mathematics courses, particularly at the higher level. The

key components of an OFA department are: a rigorous and common curriculum; a

commitment to a collective enterprise; a commitment to all students; and innovative

instructional practices (1996). These will be discussed more fully in Chapter 6.

I began to believe that the Adderley mathematics department had begun to change

and reshape itself as an OFA department. Therefore, in the second stage of data

collection, I asked questions of the teachers and observed the departmental culture and

climate, specifically investigating parallels between aspects of the Adderley mathematics

department and the key components of an OFA department. In addition, I explored the

role of leadership, and looked for historical trends that may have facilitated and

foreshadowed the direction of the changes taking place.

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CHAPTER FOUR

STAGE ONE FINDINGS

Several interesting patterns emerged from the first set of interviews and survey

data. As noted, the first stage of data affected and altered the direction of this study. I

present here the key themes that ran through the survey results and interviews that caused

this. Chapter 5 presents the findings of the second stage of interviews.

Stage One: Survey Data

The surveys I initially administered first asked the teachers for background

information, such as gender, age, and years of teaching experience. Later they were asked

to gauge the extent of their agreement with a number of statements using likert-like

choices (see Appendix A). Results of these data are shown in Table 2.

In response to the statement, "Mathematics is a good field for creative people"

(question b) the average response was 4.33. Similarly, the average response to the

statement "Mathematics education is a good field for creative people" (question c) was

4.42. Every teacher surveyed marked "Agree" or "Strongly Agree", and the mean score

reflected this general agreement.

As far as information related to the presentation of mathematical content and

classroom management strategies, responses to items g and h may also be illustrative.

With respect to these statements, 7 of 12 (58.33%) indicated disagreement or strong

disagreement with item g, and 6 of 12 (50%) indicated disagreement or strong

disagreement with the item h. The inclination in some of these teachers to change their

49

management and presentation strategies is apparent, and the interviews provide further

details regarding these changes.

Table 2

Response Frequencies of Likert Survey Data

Question Response Choices Summary Statistics

1=StronglyDisagree

2=Disagree 3=Undecided 4=Agree 5=StronglyAgree

Mean Response

StandardDeviation

a) Mathematics will change

0 0 4 5 3 3.92 0.79

b) Mathematics is creative

0 0 0 8 4 4.33 0.49

c) Mathematics education is creative

0 0 0 7 5 4.42 0.51

d) Familiar with NCTM Standards

0 0 0 5 7 4.58 0.51

e) Familiar with Illinois State Standards

0 1 2 7 2 3.83 0.83

f) Core-Plus is rigorous

0 0 4 4 4 4.00 0.85

g) Presentation will stay the same

2 5 1 3 1 2.67 1.30

h) Classroom management will stay the same

1 5 1 5 0 2.83 1.11

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Conversely, 4 of 12 teachers (33%) indicated agreement or strong agreement with

item g, and 5 of 12 (41.6%) indicated agreement or strong agreement with item h. It is

not clear whether these teachers believe that their teaching practices are already

consistent with methods recommended by the NCTM and the developers of CPMP, or if

they do not believe such changes are necessary.

Another area of interest is the teachers' familiarity with the NCTM Curriculum

and Evaluation Standards and the Illinois State Learning Standards. In response to the

survey statement, "I am familiar with the National Council of Teachers of Mathematics

Curriculum and Evaluation Standards", the average response was 4.58, where 5

represented “Strongly agree”, suggesting that the teachers felt confident in their

knowledge of the NCTM Standards. Of the 12 teachers surveyed, all indicated that they

"agree" or "strongly agree" with this statement.

This contrasts with Weiss' findings that only 56% of mathematics teachers

nationally considered themselves "well aware of" the NCTM Standards (Weiss &

Raphael, 1996). This level of familiarity with the Standards is more consistent with

Stigler and Hiebert’s findings (1999), and may partly be a result of the periodic Core-Plus

curriculum training provided by the publishers. Most on the staff had participated in this,

as is discussed below. Interestingly, although 100% of the Adderley teachers agreed that

they are familiar with the NCTM Standards, only 75% indicated agreement or strong

agreement with the statement "I am familiar with the Illinois State Learning Standards."

The average response was 3.83.

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Again, it is important to note that these are only perceptions of the staff; actual

familiarity with the Standards or other reform documents is unclear. However, it is

important to note because Weiss' study measured teachers' perceptions as well. The

difference between responses of this department and the national sample of teachers is an

indication of the potential for a reform-oriented direction.

Further in the survey, the teachers were asked to consider several specific

classroom strategies and indicate how frequently they intended to emphasize these

methods (see Appendix A). Table 3 gives the results of these data.

Table 3

Response Frequencies of Intended Instructional Practices

Question Response Choices Summary Statistics1 = Not

emphasized at all

2 = Occasionally,

but not frequently

3 = Emphasized frequently

4 = Almost always

MeanResponse

Standard deviation

a) Passive reasoning

0 9 3 0 2.25 0.45

b) Active reasoning 0 2 7 3 3.08 0.67

c) Mathematical procedures

0 5 6 1 2.67 0.65

d) Mathematical facts

0 5 7 0 2.58 0.51

e) Passive applications

0 6 5 1 2.58 0.67

f) Active applications

1 2 4 5 3.08 1.00

g) Communication 1 0 4 7 3.42 0.90

h) Passive modeling

1 7 4 0 2.25 0.62

i) Active modeling 2 1 5 4 2.92 1.08

j) Multiple representations

0 1 6 5 3.33 0.65

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Interesting results regarding the control and authority issue in the classroom

related to the teachers' perceptions of passive and active reasoning on the part of the

students, items a and b. Of those surveyed, 9 of 12 (75%) indicated that they would

emphasize passive mathematical reasoning only occasionally. Similarly, 10 of the 12

teachers (83.3%) intended to emphasize active reasoning frequently or almost always.

In addition, intended use of both passive and active mathematical modeling is

indicated in items h and I, respectively. 8 of the 12 teachers (66.67%) indicated that they

would not emphasize passive modeling at all, or infrequently. On the other hand, 9 of the

12 surveyed (75%) intended to emphasize active modeling on the part of the students

frequently, if not always.

The data further show that 11 of the 12 (91.67%) intended to emphasize

mathematical communication, or asking the students to speak and/or write clearly about

mathematical ideas, frequently or almost always. These results suggest that the

department agrees, at least in principle, with one of the central tenets of the Core-Plus

curriculum: active learning (Coxford et al., 1998).

Yet later in the survey, when asked to judge their expertise in using writing in

their classes, none rated themselves as "expert", given the choice between "novice",

"intermediate" or "expert". Furthermore, only three of the 12 (25%) considered

themselves experts in using group work in their classes. This is consistent with the

portrayal of a department that is still learning how to use various instructional methods.

They may not be confident with the means to some of the larger ends of a reform-

oriented curriculum, such as developing the "network of knowledge" held by individual

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students, as well as nurturing a supportive mathematical community (National Council of

Teachers of Mathematics, 1998, p. 34).

Survey Data: Teacher Interviews

Each teacher was given a list of metaphors for the teaching of mathematics, and

was asked to select the three that best characterized his or her beliefs before each

interview began. Later, they were asked to explain these choices. Figure 2 summarizes

these results, and one can see that the most-selected choice was "coach", followed by

"orchestra conductor" and "entertainer".

Sizer (1992) presents coaching as an appropriate metaphor for the teaching that

should occur, but often does not, in many schools. Echoing recommendations made in

mathematics reform documents, Sizer suggests that the process of teaching and learning

is dependent on active student participation, rather than teacher exposition. He compares

this process to coaching a javelin thrower when he writes, “You throw. I criticize,

suggesting some possible improvements. You throw again. And again I criticize. This is

how skills in a strong athletic program are shaped. The analogies to intellectual training

are powerful and apt” (p. 106).

Similarly, one Adderley teacher noted the following regarding teaching the new

mathematics curriculum:

Well, you're trying to put people in a very rich environment, putting problems and obstacles around them, and trying to get them to deal with those problems. You're trying to coach them in ways that they might approach it, and talk to them about what they're trying to do.

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Insert graphics here

Figure 2

55

The choice of "Orchestra Conductor", was interesting, because it reflected a sense

that part of the teacher's job is to keep the class running smoothly and create harmony out

of 30 different voices in the classroom. However, the implications for authority relations

could depend on what one considers the role of conductor. One might see this as a

facilitating role, where the teacher encourages the best performance from each student.

On the other hand, orchestra conductors have been characterized periodically as despotic

and heavy-handed. The following comments support the former interpretation:

Especially in the Core Plus program, where your job as a facilitator is to keep 30 kids working smoothly and cooperatively, it's as difficult or more difficult than an orchestra conductor. In an orchestra, everybody already knows what they want to do, and you have to get everybody going in the same direction and keep them going in that direction.

The teachers were also asked to choose from a similar set of metaphors to

characterize their beliefs about mathematics learning. Overwhelmingly, the most popular

choice was "Working a jigsaw puzzle", followed by "Cooking with a recipe" and

"Building a house," as shown in Figure 3.

In comparing mathematics learning to a jigsaw puzzle, one teacher

remarked,

It's a puzzle because, um, and this is the hardest thing to convey to kids, if you step back and look at what you've learned in all of your math classes, and I didn't get it until I was in college, you can kind of understand what you did after you step back and take a look at everything together. Everything kind of fits together in a whole if you step back.

Although the choices of "building a house" and "cooking from a recipe" might be

interpreted as viewing mathematics learning as rote, according to scripted plans, other

themes emerged from these teachers' explanations. First, these teachers were product-

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Insert graphics here

Figure 3.

57

oriented: the end result of mathematics learning should be as useful and real as a house or

a nourishing meal. Second, the teachers noted the artistic and creative aspects of building

and cooking as analogous to mathematics learning. This is reflected in the following

teacher's comments regarding mathematics as a field of study, in which she compared

mathematics to a blueprint and a cookbook:

Mathematics is dynamic...I guess I could say it's a blueprint because you have a structure you're beginning with. But when I think of a blueprint, I think about a house, and the house goes up...but even if it was the same house from person to person, the interiors change and the exteriors change and the design is changed, and there's some flexibility there… And then there's a wall that will be broken down later and the blueprint changes a little bit, so I guess I could say that mathematics is a little like a blueprint to some degree. ...A cookbook too, in that same respect because it does start with so many ingredients. But each cook adds their own flavor, their own personality, their own individualism to whatever they're cooking. And also, cooking is exceptionally creative, and I actually believe that mathematics is very creative, too.

I further posed a number of open-ended questions related to perceptions of

authority. Specifically, I asked, "If someone were to say that you were 'in control' of your

classroom', what would that mean to you?" Most teachers noted that control meant to

them a civil environment, or at least the lack of total chaos. This base definition was often

characterized by statements such as, “students not running around the room”, or “no

objects are being thrown around.”

Yet, deeper interpretations of "control" seemed to have culturally taken-as-shared

qualities that signaled a unique receptivity to this style of teaching within this department.

In nearly every case, these teachers used vocabulary that demonstrated the interpretation

of "authority" and "control" to mean being able to step back and let the students explore.

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Although paradoxical, the teachers make some intriguing statements about this tendency

to "control by letting go". Some illustrative comments were:

Control means that I've set it up well enough for the students to be on their own. That I've designed the lesson well enough that I can let go. I can literally walk out of the room and things are still be going on the way they should be going.

The old teaching style used be that you were in control if no one else was talking except for you. And to this day, if I'm introducing something, I do expect the students, if not listening, at least not talking... But being in control now to me just means that the students are actively involved in doing math, and thy can be talking about the math, they can be, like, measuring each other, or whatever the situation calls for. But if an administrator would walk in and see a class in disarray, and heaven forbid people laughing or enjoying themselves, I think they need to know that actual math is going on, learning is taking place, even if the room is not quiet. Which is the way it used to have to be.

In the olden days, if people were quiet, it didn't matter if you had them (actively engaged) or not. If their eyes were open and they were facing you, you knew that you had control of the class. It's not really that the teacher has control in the class now. Like, it used to be that the teacher had to be totally in charge. I don't feel like I have to be totally in charge anymore. I obviously am the more educated person in the room, but if I can get around to each group and see that they're actively working then that's good. If they're not I need to coerce them somehow to get busy or get on task.

Control...hmm...not a word I use a lot. Control could be that the class is functioning, the students are dealing with some human activity that is an outgrowth of the course we teach, and that they're involved in working on that problem.

Reflections on the Data

These teachers' responses have a number of interesting implications. Most

significantly, it appears that this department does not put a high value on authority in the

traditional sense of the teacher being the source of all knowledge. In particular, a majority

of the teachers expressed a philosophy consistent with the NCTM Standards, and other

59

reform documents. That is, most suggested that they would take a student-centered

approach, emphasizing collaboration and mathematical communication.

In addition, their educational philosophies seemed generally at odds with an

"absolutist" view of mathematics education. As previously noted, I was openly surprised

by these responses in light of my history with the department. In sum, they suggested that

the department may have been particularly receptive to other reform-oriented

recommendations, and perhaps to this curricular change overall. I suspected there were

deeper issues to be uncovered.

Of course, these conclusions are based on the subjective words of the

teachers, and lack the corroborating evidence of classroom observation. Still,

these are significant, as they are indicative of the culture of the department,

revealing beliefs, values, and philosophies. As Fullan and Stiegelbauer (1991)

note, "Ultimately the transformation of subjective realities is the essence of

change" (p. 36).

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CHAPTER FIVE

STAGE TWO FINDINGS

As a result of my interpretations of the first stage of data, I visualized

obtaining a wider picture, a "snapshot" of the department for the second stage of

data. Rather than focusing on specific aspects of classroom dynamics, such as

perceptions of authority, I sought the teachers' perceptions of the overall cultural

context. Weissglass (1992) notes,

One advantage of situating change efforts in a cultural context is that it enables educators to address the complex psychological and political reality of schools and classrooms, rather than focus on the more technical issues (new methods of assessment, curricular or achievement standards, and technology, for example). (p. 196)

During the Stage two interviews, I first asked the teachers their overall

impressions of the department and curricular materials. In general, I asked their

perceptions regarding the new curriculum, the reasons it was implemented and its

appropriateness, and about collegiality and professional development in the department.

Furthermore, I asked them to describe their perceptions in instructional practices and

whether the new curriculum facilitated a commitment to all students at Adderley High

Schools. Again, all of these subjects relate to elements of the OFA framework, and I

asked the teachers to describe any changes in these they may have noted. In addition, I

asked them about the role of leadership in the department, perhaps a key variable in

implementation of the NCTM Standards (Garet & Mills, 1995).

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Significant Cultural Traits

Guided by the definition of culture offered by Weissglass (1992), I interviewed

the teachers, seeking information related to their overall attitudes, beliefs, values and

practices. The teachers were asked first about their attitudes toward the new curriculum.

More specifically, I asked if they believed that the CPMP/Accelerated Mathematics

hybrid was working. Overall, there was general agreement that this program seemed to be

the right direction for the department. Every single Core-Plus teacher interviewed had

positive things to say and most seemed optimistic about the program's future success.

Some examples are:

I’m really high on the Core-Plus, and my reasons are many. I've been around since the time when teachers just lectured to the students for 50 minutes …Core-Plus is excellent in getting them to not only participate in their own education, which I think is the biggest factor, but get to do some critical thinking, and work with another person…The one negative thing is that, I think we're presupposing that kids come into high school with a lot of algebra talent.

…(The Core-Plus students') understanding of slope, their understanding of graphing of a table, their understanding of how the numbers in those tables relate to a story problem is phenomenally better than what is see the (non Core-Plus) juniors doing. But if I ask them to do polynomial work, or exponent work, it's like 'Ack!' They don’t know what to do.

It’s not perfect yet, but it’s better (than the traditional curriculum). Because it’s our first year, we’ll need to sit down at the end of the year and decide what worked and what didn’t.

Core plus, is not only exciting, it works if we take out the piddly stuff and try to figure out what were trying to develop over a 3-4 year period of time.…read, interpret, make sense out of things and have kids really engaged in what they're doing.

The kids enjoyed the class because it dealt with real life things. They learned more in this year than in any other years because there was less emphasis on taking tests, and more emphasis on thinking communicating

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with another person and not being afraid to answer the question because it's what you think (that's important), not what you do.

The staff is buying into the program for a lot of reasons. Deep down, most of us feel that it’s probably the way to go, but we’re struggling with some of our own insecurities because even some of our young teachers were not trained to teach this way.

Despite this general positive feeling, it is important to note that each teacher

above cited specific concerns, including the department chair who commented,

I think Core Plus, what that did, is put us in the right century. The idea that math being only 'equations', that's gone…. We're at a place now where I think we should be. We're done selling it- it's there. It's my job now to critique it. I don't want people to start thinking that because it's there that it's golden.

This cautious optimism was echoed in every teacher’s comments, although several of

the staff expressed their anxieties more bluntly:

Right now, it's pretty scary. Because we're out alone, and our hearts are not totally there. Because even if what you had wasn't working, it's comfortable. This is not comfortable. This is different.

The program is very time consuming for some teachers who don’t know how to handle it. I've seen three teachers who have been overwhelmed by it. And these are veteran teachers. I don't know what to tell them. You know, 'Just relax, don't get so uptight', but in the same sense, how? Everyone's on different paths, trying different things, and we don’t know what works.

I’ve been teaching 29 years and this is the worst year I’ve ever taught, because the stress level is unbelievable. I have huge classes….It's a major transition, it's a brand new course, with a lot of new technology, and there’s no time to get good with anything…There were just so many things all at once. It has not been a pleasant year. If I don’t teach this again next year, someone's going to die, because I've put in a whole lot of work.

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These comments are indicative of the numerous anxieties being negotiated within the

department as they learn about this curriculum. It is important to note that the Core-Plus

program is designed as a three-year course for all students (Schoen et al., 1998). The

Adderley mathematics department has not yet implemented the third year of the program,

and only began widespread implementation of the second year in 1999. Many teachers

have only seen Year 1 or Year 2, and lack the benefit of perspective over the whole

course sequence. So, some of these uncertainties are understandable and probably natural.

As Lortie (1975) writes, "education is a tenuous, uncertain affair. It is necessary to keep

such uncertainty in mind if we are to understand the psychic world of classroom teachers,

for uncertainty is the lot of those who teach" (p. 133).

So it would seem that the department is at a critical juncture. While many of the

teachers are optimistic, they are mostly unsure where the program will lead. As Fullan

and Stiegelbauer suggest, "Change is full of paradoxes" (1991, p.102). Certainly, this

department might empathize with this sentiment, as the second stage interviews revealed

paradoxes, tensions, and contradictions. Some of the most common themes expressed by

these teachers are discussed below.

Skills and Core-Plus

The debate over the emphasis on mathematical skills in the Core-Plus sequence,

or lack thereof, has had particularly far-reaching results in the implementation of the

program. Selected comments illustrate this:

64

I think there needs to be more of some old traditional-type stuff, as long as colleges are still requiring that they have the basic principles of algebra and geometry.

I am truly concerned that (the students) are getting enough math. There are so many topics that I feel that we don't get to.

The more I teach it, the more I feel that there’s stuff that’s missing, but I feel that they're giving the kids challenges with different kinds of material. I guess I'm torn between this and a more traditional “skill building” course.

My key word this year has been 'balance'….so I'm trying to build some organized chaos as far as getting them to have some of the basic skills in algebra, and yet apply that, and that's the hard thing.

As noted, the Adderley mathematics department has not yet fully implemented

every course in the Core-Plus sequence. Therefore, the debate over the rigor of the

curriculum, and whether CPMP adequately stresses algebraic skills may stem from this

uncertainty. While algebraic skills are a central part of most first year algebra courses,

they are not particularly emphasized in the Core-Plus Year 1 or Year 2 curricula. For

example, students solve equations using the graphing calculator, but there is relatively

little algebraic manipulation. Because CPMP is designed around "strands", algebraic

skills are revisited, increasing in depth and rigor each year. Therefore, many of the

elements that would be crucial to a high school freshman in a traditional course are

indeed present in the third and fourth years of the Core-Plus sequence.

In fact, some research suggests that students having passed through the full CPMP

program score as well or higher than traditionally taught students on nationally known

standardized tests. (Schoen et al, 1998). In particular, this research studied student

performance on the Ability to Do Quantitative Thinking (ATDQT), which is the

mathematics subtest of the Iowa Tests of Educational Development (ITED), and a test

65

styled after the National Assessment of Educational Progress (NAEP) (Schoen et al.,

1998). Because algebraic skills are assessed as part of these tests, the concerns expressed

by some the Adderley teachers may be premature.

Yet, without having examined Core-Plus Years 3 and 4, some Adderley teachers

still perceived a lack of emphasis on mathematical skills. For example, nearly every

teacher interviewed cited “solving for x” as symbolic of this debate. Referring to students'

ability to use an algebraic algorithm to isolate a variable in an equation, the mantra

throughout the department was, "When will they learn to solve an equation?" Coming

from some teachers, this question reflected sincere concern about the rigor of the

program. From others, this phrase was inflected with exasperation and sarcasm. The real

question for the latter group was, "When can we get past worrying about 'solving for x'

and move on to more important issues?" One teacher noted,

There’s still too large of a percentage of people (in the department) who are locked into 'we have to teach (students) how to solve an equation', and order of operations and things like that. Look, if we can replace skills with manipulating (graphing calculator) screens, and not be afraid to make a few mistakes, and teach them to solve in a multiple of ways, we’re doing a better job at educating them.

The Honors Level

The debate over "skills" affects other aspects of the departmental culture as well.

Specifically, the teachers have begun a dialogue centered on the appropriateness of these

materials for the entire student body. The different viewpoints are exemplified by these

comments:

I also don't know if Core-Plus is the type of class...I mean, we're making everyone take it. There's some stuff in there that's deep. Maybe it's just for the honors-type kid. I don't know.

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At this point, it’s going fine. I'm just concerned. And because I'm not familiar with all 4 years, and we haven't taken a group (of students) through all four years, I don't know if they're getting enough math. I'm just concerned that at the honors level…that maybe they're not getting exposed to enough topics.

We do have a (college-intending) population, and those are the people I'm concerned about.... I'm not comfortable doing Core-Plus to the extent that were doing it with the honors-type kids…. I'm feeling that the Core-Plus isn't tough enough in the abstract area, and they're not going to be able to compete….I've always been leery about this top group.

As a result of my personal experience with this department, and the historical

emphasis on classifying students as "honors" or "non-honors", I was not surprised that

this is an unresolved issue. Of course, serious concern about the honors level student is

appropriate. As a result, the curriculum designers have suggested that the fourth year

course under development is intended for preparation of students for college mathematics

(Schoen et al., 1998).

Perhaps the tension is rooted in disagreement between what "honors" students

should be able to do, versus what all students should be able to do. In other words, is

Core Plus ideally suited for honors students because of the deep conceptual content? Or,

is it inappropriate for these college-intending students because it does not stress skills in

the first two years? One wonders whether the definition of "rigorous" mathematics will

similarly need to be redefined. Is "rigor" synonymous only with advanced symbolic

manipulation, or in a reform-oriented environment, does it mean more than mastering

these skills? The department will ultimately have to resolve these issues.

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The Accelerated Mathematics Program

The perceived weakness of the Core-Plus in emphasizing and reinforcing

mathematical skills has had another result discussed at length in this paper: the

implementation of the Accelerated Mathematics Program. The debate in the department

centers on whether Ack Math fills holes in the Core-Plus program, or actually

undermines it. A cognitive psychologist might argue the latter.

The Accelerated Mathematics Program is essentially based on behaviorist

psychology. It is a high-tech version of Skinnerian programmed instruction, under which

knowledge is decomposed into small bits to be mastered individually in a “drill and

practice” format. Much to the chagrin of reform-minded mathematics educators, drill and

practice has had significant impact on American mathematics education since E.L.

Thorndike published The Psychology of Arithmetic in 1922 (Schoenfeld, 1987).

In Ack Math, the small bits of knowledge are presented as multiple choice

worksheets printed by the computer, which are to be completed by the students (see

Appendix C). These problems are similar to those commonly used in standardized tests,

which emphasize predominantly low-level skills (Lacampagne, 1993). Students receive

the positive reinforcement of meeting an objective and having the objective "checked off"

next to the student's name on the computer. One might argue that instructional methods

based on this theory conflict with the emphasis on active learning and constructivism, a

theory that permeates the NCTM Standards, and the Core-Plus curriculum. The NCTM

(1998) alludes to constructivism in the following passage:

Some lines of research suggest that people do not generally learn concepts by building up pieces of knowledge…Instead, they are more likely to plunge headlong into a problem situation, bringing whatever facts, procedures, and understandings they have at hand. Thus, expecting

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students to master as set of so-called prerequisites in a prescribed order before they attempt to tackle challenging problems in a prescribed order is not necessarily productive. Although students taught in this manner may learn facts or procedures well, they often are not sure when or how to apply them, and they may also forget them quickly. More importantly, these students have little chance to experience satisfaction that comes from grappling with complex and interesting tasks. (p. 34)

Certainly, the NCTM is quick to note that the learning of routine skills and facts

has an important place in mathematics education. Yet, they also advocate a coherent

curriculum, which will be visible in the "consistency of style and approach" (p.29). The

issue of consistency seems to be the most significant for the Adderley teachers. These

comments illustrate both sides of the issue:

Basically, what I'm hearing, is that we're not really doing Core-Plus. What we're doing is tinkering with a little bit of this and a little bit of that, meaning the Accelerated Math Component.

To me the essence of whether (the students) understood anything is, are they able to apply their knowledge in a situation they’ve never seen? The more we use the Ack Math, the less that happens...Core math is a great model. It can be supplemented with Ack Math. The biggest interference with the program is the ACT-type question. When we continue to say that ACT scores are important politically, then you have to have those (Ack math) questions in there.

Ack math makes the job of recording what (the students) do very easy. It is much easier for the instructor, but we miss the essential element in education, in my opinion, and that is to read, interpret, make sense out of things, and have kids really engaged in what they're doing. For me to engage them (in an Accelerated Mathematics format), I have to come in with things like, 'Well you have to learn to read a diagram in market, because that's what happens when you look at a blueprint'. But I never bring in a blueprint in, I just bring in these asinine worksheets, and they do their little manipulations, and it just seems to me to be to be a waste of time.

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Other criticisms of the program have been that the objectives are often not aligned

well with the Core -Plus topics. In particular, because the CPMP Geometry strand

stresses different topics than a traditional Geometry course, the department is having

trouble aligning the Ack Math objectives with the Core-Plus topics. The department plans

to correct this problem during the summer of 2000, but this has caused some problems for

the teachers trying to present a coherent sequence of lessons. Furthermore, the technology

used to connect a printer, scanner, and laptop computer has been problematic at times; a

few teachers have experienced problems printing and accessing computer files easily.

On the other side of the issue, some teachers have found that students seem to be

fond of Ack Math. Despite the problems with the Geometry strand, they suggest that it

supplements the Core-Plus materials well, and that students have enjoyed the change of

pace it represents from the Core-Plus instruction.

I like the Ack Math, because it nicely complements the Core math. My students love it. On the couple of times that we haven’t been able to do it, my students have been disappointed. They like the flexibility of it, they like the individualness of it, they like the immediate feedback of it. I like the immediate feedback of it. It gives me time to work with individual students. I feel like I’ve been working with individual students more

I would say that the Ack math at least made classes not teacher centered. It made implementing Core-Plus easier. Just having kids walking around is mind-blowing for some teachers. Now you'll see lectures that are smaller and shorter. People are trying different things, and have a comfort level with the technology. You'll see more in groups and you'll see more conversations between students. It's a much more humanized classroom.

It is important to note that the last comment was made by the department chair,

who has evidently taken the position that Ack Math was significant in "softening the

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blow" to some teachers. He contends that the program is certainly important in

reinforcing basic skills the students may need on standardized tests, or in college-level

mathematics. Yet, the reshaping of classroom interactions to include group work and

student discourse has also been necessary. In his opinion, Ack Math has been a means to

that end.

The Bipolar Department

Earlier, I alluded to a tension I perceived during my teaching tenure at Adderley.

That is, I suggested that the department was split almost evenly between younger teachers

with 10 years or less of experience and veteran teachers who have been teaching 20 years

or more. In fact, Figure 4 illustrates that, in 1999-2000, there were 11 of these "younger"

teachers, and 10 "veteran" teachers. I also suggested previously that significant

differences existed between the philosophies of the older teachers and younger teachers

in the department.

This issue is still prevalent at Adderley, but attitudes toward reform are not so

easily categorized as to neatly fall along the lines of “young” versus “old.”. For example,

when I asked the department chair why he thought the department had been mostly

receptive to the new curriculum, he noted, "A couple things were going on. You had

retirement, and a lot of new people coming in at the same time, and enough senior faculty

who knew what they were doing wasn't working". In other words, it took a mixture of

youth and experience to set the stage for the new program, and no one faction specifically

championed it or took arms against it.

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Insert graphics hereFigure 4: Years of experience

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However, the teachers themselves have conflicting theories regarding the role of

experience in the department and how this variable is affecting the departmental culture.

To some, the veterans are standing in the way of reform, while others believe that the

veterans are actually at the heart of reform. Furthermore, others believe that the younger

teachers are struggling as much as the veterans, rather than driving the changes. These

contradictions are evident in the following comments:

The Core takes us closer in terms of where we want to go in terms of application, analysis and reasoning. We're looking for things that work, that get kids excited about mathematics, and we're dragging along some of the ones that…er... there's been enough changeover in staff that that's become more of what's expected. But there are always the ones that say they’re doing stuff, but go about their same old ways, and go about their 'dittoes', so to speak.

I think that a lot of our younger teachers are more into the working around with the groups rather than standing up there spewing out information, and I think that's very good. They're very computer-literate. They didn't even have computer science majors when I went to college...I think the younger teachers, that's where it's at. Some of us are going to be leaving in the next five years, and we're going to be left with all young teachers.

The teachers that love it have been around a lot longer, and they've seen that the (traditional) stuff just isn’t working. A change needs to be made. (Traditional mathematics) is maybe too dry and boring, and the Core-Plus has a little more spice and real world applications. The ones that hate it, they're also the ones that have been teaching a long time, and love the old stuff and hate to see the change.

The old fogies are set in their ways. They have a to radically change. They’re going along with it, but they're still stuck in their ruts…and I don’t know if they’re willing to give it a chance. I'm not sure if younger teachers are buying into it, either. They're a little 'iffy'.

From these comments, it seems that the department perceives that both younger

and older teachers are actively supporting the reforms, and taking leadership positions in

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the implementation process. In addition, the department has subsets of both groups who

are particularly uncomfortable with the CPMP/Ack Math program, and are moving more

slowly toward compliance with the new departmental standards.

With respect to attitudes toward the new program, perhaps a more appropriate

classification system would not divide the department between "younger" and "older"

teachers. Instead, one might argue that the department consists of some "stabilizers" and

some "visionaries", categories from some personality type research. Again, stabilizers

conserve the traditional values of the institutions in which they work, and visionaries are

inclined to lead rapid and dynamic change (Clark & Guest, 1995).

However, I do not wish to imply that the stabilizers in the department are

uniformly opposed to reforms, and are somehow sabotaging the change effort. As one

teacher suggested, "We certainly have some teachers that aren’t as open to change.

However, I don’t think that we have any that are completely closed to change. They’re all

willing to at least give something a try." Of course, teachers, like anyone else, do not

learn or accept all new ideas all at once or at the same rate. Because these teachers have

moved much more cautiously toward new methods, they may act as a check against

change in the department proceeding too rapidly.

Leadership

Despite these tensions and contradictions, I argue that the department has

generally accepted the new program, and is in the process of implementing it

successfully. Certainly, the faculty deserves much of the credit for this. As suggested by

the department chair, an influx of younger teachers and a core group of veterans open to

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change set the stage for the process to begin. Since then, these teachers make it very clear

that the entire department has worked diligently to make these reforms successful.

However, I argue that leadership, particularly from within the department, has been the

key element to this receptivity.

Upper Level Leadership

As background information, it may be important to note that Adderley High

Schools saw significant changes in leadership following the 1998-1999 school year, the

result of the respective retirement and departure of the previous Superintendent and

Director of Curriculum. The new Superintendent moved into her position after serving as

Assistant Superintendent for several years, and the new Director of Curriculum was hired

from outside the district. The Principal at West Adderley High School then became the

Assistant Superintendent, and the new Principal was also hired from outside the district.

Despite so many changes, this upper level administration has been crucial to the

implementation of the new mathematics program. From all accounts, the mathematics

department has been given a measure of autonomy and license to take risks. In addition,

the reform efforts have been supported financially. For example, the technology

necessitated by the Accelerated Mathematics Program requires a strong financial

commitment, as does the support of continued teacher training. Furthermore, the

department chair has noted a willingness to support his efforts to use standardized test

data and student enrollment data to assess the new program's effectiveness. This

sentiment was not as prevalent under the previous administration.

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As one teacher suggested, "If it wasn’t for administration support and willingness

to take some risks, this never would have happened". Furthermore, the department chair

said, "The new administration has been super. I mean, it’s our ballgame. We don't have

people looking over our shoulder. They'll call us in and ask: what are you doing, how are

you going to do it, and CAN you do it? That's the big question."

Departmental Leadership

As clearly crucial as the upper level administration has been, leadership from

within the department has arguably been an invaluable part of the new mathematics

program. As proposed by Fullan and Stiegelbauer, six interconnected themes related to

successful reform efforts may illustrate the educational leader's role in this process. These

are: vision building, restructuring, staff development and resource assistance, monitoring

and problem coping, evolutionary planning, and initiative taking and empowerment

(1991). Although Fullan and Stiegelbauer discuss these in the context of systemic school-

wide change, these themes still have relevance at the department level. The remainder of

this chapter discusses the department chair's role in the implementation of the Core-

Plus/Ack Math hybrid in relation to these themes.

It must be noted, however, that my data do not support a comprehensive and

detailed analysis of these themes. The examples and illustrations offered here are

presented only as indicators of the significant effect the department chair has had on the

implementation process.

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Vision Building

According to Fullan and Stiegelbauer, vision building "permeates the organization

with values, purpose, and integrity for both the what and the how of improvement"

(p.81). This involves providing direction and driving power for change, as well as

facilitating a shared vision of the strategy for implementation. The following comments

are indicative of the teachers' attitudes toward Mr. Blakey and his efforts to build vision

within the department and among other administrators:

This guy is working. He keeps himself informed, he talks to other people, and he’s got administration that is supporting him... And he is so sure this is the way we need to go. And he's willing to be the forefront leader, he's taken all the classes, he's a year ahead of us, so he can help us. I think that's the big thing: we've had a little more support. (Blakey) is trying to get us to see the long picture of things: to see if we're building a way of thinking (in students), rather than just doing.

The department chair leadership is huge! This program is working because the leader believes in it enough, and the leader has convinced administration that this is what should be done.

Restructuring

This theme refers to the leader's role in shaping the school as a workplace,

shaping the policies that affect time for team planning, staff development, and joint

teaching arrangements. The department chair may have the most impact on these issues

through scheduling teaching assignments. To illustrate this, the proposed course offerings

of three consecutive school years are shown in Appendix D. The differences from year to

year result from the Core-Plus/Ack Math program being phased in, and the traditional

courses being phased out. By the 2001-2002 school year, every course other then

Advanced Placement classes will be "Integrated", or a Core-Plus/Ack Math offering.

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In addition, as the years progress, there are fewer courses offered, and the overall

structure becomes less stratified. Significantly, the number of “tracks” is reduced by one

by 2001-2002. At this time, Adderley High School students will be divided into “College

Prep” and “Accelerated College Prep” groups, rather than “Skills”, “Regular”, and

“Honors.”

As a result of this new curriculum structure, Mr. Blakey was able to design the

schedule so that "two preps is the norm". That is, teachers are in class five periods during

the day, but now only prepare for two separate courses. Because the department offered

more courses to students under the old curriculum structure, three preps was the norm at

that time. This curriculum streamlining affords teachers more time to plan and reflect on

their practice. Furthermore, more teachers have common preps, which has increased

opportunities for collaboration and shared planning. This will be discussed further below.

Staff Development/Resource Assistance

Certainly, collaboration relates to the staff development aspect of the

implementation process as well. Because the curriculum is now less differentiated

between student tracks, the teachers have greater opportunity to collaborate. For example,

all freshman classes are using the same text; the only differences between classes are the

pace and the depth of coverage. So, rather than being divided among three separate texts

and three separate sets of goals, all teachers of freshman students are on the same “sheet

of music". One teacher noted,

The biggest difference that I see is, several years ago, everybody was an island, where everybody did their things their own way. Certain people talked because they chose to. Now there’s a standard for everybody. Everybody has a standard. We only have 2 classes for freshmen, instead of

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5 or 6. When we talk at inservice, it applies to everybody. Consequently, there’s a lot more sharing. I see standards for every teacher, every class. I feel like it’s a real department now.

Another significant example of the support for staff development is the

department's ongoing Core-Plus curriculum training. Every summer, the department

encourages teachers to attend weeklong training sessions conducted by the CPMP

designers; each session focuses on a specific year of the program. Some of these sessions

are held in Michigan, so teachers stay in hotels and attend training for several days.

During the summer of 1998, the summer after the pilot program, the department

sent 14 teachers to Kalamazoo for this type of extended training. For many, it was their

first extended interaction with the course materials, and this period may have been a

significant factor in the department’s receptivity to this program. One teacher observed,

The one thing that changed everybody’s mind was the trip to Kalamazoo. We had 14 teachers, more than half completely jazzed (after the training). There were those that were more reluctant to change, but they walked out saying, ‘Wow! This might work, but I need more training’. (Mr. Blakey) scheduled it so that, for those people, their prep period was when they could go in and sit and visit a class. And some people did that.

The ongoing training, and the ability to observe each others’ classes are

indications that staff development is taken seriously at Adderley, and is important to the

overall implementation process. While this is a staff development issue, it also seems to

cross over into vision building. Fullan and Stiegelbauer write,

Implementation, whether it is voluntary or imposed, is nothing other than the process of learning something new. One foundation of learning something new is interaction (emphases in original). Learning by doing, concrete role models, meetings with resource consultants and fellow implementers, practice of the behavior, and the fits and starts of

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cumulative, ambivalent, gradual self-confidence all constitute a process of coming to see the meaning of the change more clearly. (1991, p. 85)

Also significant has been the resource assistance, particularly in light of the

technological requirements of the Accelerated Mathematics Program. As noted,

supplying teachers with laptop computers, printers, and scanners requires a great

financial commitment of the part of the district. Furthermore, it necessitates technical

support. Although there have been some minor difficulties with this technology in

isolated cases, it seems to be working well for the department. One teacher expressed his

approval with this support by saying,

Blakey is fantastic. He’s got so many great ideas, and he’s getting them done. For example, he wants to do Accelerated Math. Well, that means that all the teachers who are doing Ack math need their own laptops. Done. Got it. “Oh wait! Now we need special printers”. Got it.

Monitoring/Problem Coping

Monitoring and problem coping are particularly complex issues as defined by

Fullan and Stiegelbauer (1991). Monitoring provides access to good ideas and innovative

instructional practices, and it subjects ideas to scrutiny, which helps to weed out mistakes

and develop more promising practices. Certainly, these teachers have no qualms about

subjecting Mr. Blakey's ideas to scrutiny. They are consistently asking questions, and

presenting alternative ideas to him, as I was told on several occasions.

Encouraging staff to visit each others’ classrooms is evidence of healthy

monitoring, as is the overall collegiality in the department, although there is some

disagreement as to how the new curriculum has impacted this.

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Some teachers characterized the East Adderley teachers as historically more

willing to collaborate, and the West Adderley staff as more insular. However, this may be

changing.

Some comments noted from East Adderley teachers were:

I haven’t gotten to meet with other teachers lately. I kind of feel like I’m out of the loop.

I don’t collaborate with other teachers. There are lots of little cliques.” There's tremendous willingness to work together here… I see a lot

more independence now, particularly with the younger teachers. I don't know if they sit down with the same amount of collegiality that we used to.

Some comments from West Adderley teachers were:

As faculty, we are starting to talk a little more. We’re starting to help each other a little more.

(Now) there’s a lot more sharing going on here. There is some collaboration here, although not as much as I'd like to

see. There’s still camaraderie, but not as much as at East.

Reasons for this are not clear. Again, the district has seen a number of changes in

personnel, and the chemistry of the staff at each campus may be a significant contributor

to this.

Another important aspect of monitoring is collecting data in order to assess the

program. As noted, the new upper level administration, and the Director of Curriculum in

particular, is working with the department chair to gather data to assess the new

program’s effectiveness. Mr. Blakey is preparing a “Math Report Card” to measure the

strengths and weakness of the new course offerings. This report will compare statistics

for five consecutive school years in each of the following areas:

Number of students who have failed mathematics courses.

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Percent of student population enrolled in mathematics course.

Advanced Placement (AP) test results.

10th grade students’ Illinois Standards Achievement Test (ISAT) scores.

It should be noted that the ISAT is a statewide standardized test of reading,

writing and mathematics administered to students in 3rd, 5th, 8th and 10th grades.

Results are reported according to four performance levels: Academic Warning, Below

Standards, Meets Standards, Exceeds Standards (Illinois State Board of Education, 2000).

Furthermore, the district plans to compare performance of 8th grade students

according to middle/junior high school attended. Because students from five different

feeder schools attend the East campus, and students from three feeder schools attend the

West campus, there has long been concern about the variability in students’ mathematical

backgrounds. That is, students from particular feeder schools seem to have more

difficulty in high school mathematics classes than others, and the district will attempt to

gather data to support this hypothesis.

Evolutionary Planning

Evolutionary planning relates to adapting the plans to improve the fit between

school conditions and the reform initiatives. The department has demonstrated an ability

to adapt to challenges, as evidenced by the reaction to problems with the second year

curriculum. Specifically, the faculty perceived "major problems" with the Geometry

strand of Core-Plus. A number of teachers of CPMP Year 2 held a planning meeting with

Mr. Blakey to discuss these issues, and he said that the direction of the second year

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curriculum "turned on a dime" as a result. At this meeting, the decision was made to

divert from the Core-Plus sequence in favor of teaching more traditional Geometry topics

for one quarter, and these changes were implemented the day after the meeting. Although

these changes are considered temporary, the resolution of this situation, as well as the

adoption of Ack Math, shows a willingness to adapt the program to the specific needs of

the district.

One issue to consider related to evolutionary planning will be discussed further in

the next section, which discusses Initiative Taking and Empowerment. Fullan and

Stiegelbauer suggest that a blend of top-down and bottom-up participation is common in

successful reforms, and this seems to be one source of tension related to the leadership of

the department.

Initiative Taking and Empowerment

Initiative taking and empowerment involves "power sharing”, delegation of

authority, and developing collaborative work cultures " (Fullan & Stiegelbauer, 1991, p.

83). Although most of the implementation process seems to be in alignment with Fullan’s

suggestions, some tensions should be noted here. Specifically, some in the department

feel that Mr. Blakey's leadership style is primarily top-down, and that there is a need to

encourage more ownership by the teachers in the program, as illustrated by the following

comments:

I know I was not part of the decision making process, so…I don't know where (the curriculum) came from. I can guess, but I’m not sure….

I am seeing a lot of top-down decisions. I understand where (Mr. Blakey) is coming from, trying to find a solution to the detracking

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situation, and you know…I feel like these are all (Blakey’s) decisions. I see him as the “benevolent dictator”. I don't think he's trying to make our lives miserable…. I think his decisions, though, are creating chaos. And I don’t think there’s an ownership going on, like we've said 'We want this'.

However, another teacher flatly challenged this attitude, saying, "(Blakey) is not

jamming instructional styles down people’s throats, although there are a number of

faculty that feel that way." One specific issue seemed to exemplify this tension, and was

raised by a number of teachers. Mr. Blakey, in collaboration with several other teachers,

writes a weekly syllabus for each Core-Plus class, and distributes it to the staff (see

Appendix E). According to Mr. Blakey, the syllabus is meant less as a mandate than as a

guideline, providing the teachers with some sense of the pace of the course and giving

them ideas about particular problems to try with students. "I keep telling (the teachers)

that the syllabus isn’t a mandate", he said, "They should follow the sequence, if not the

timeline. I think we've got reasonable compliance, except being on time, which isn’t the

important thing."

However, there were contradictory perceptions of this issue:

Hmm...the syllabus: is it a general guideline, or is it a mandate? I wish I had more input as to what I do. I mean, I feel what I’m doing is working. If there were more weight placed on what I say, I’d say more.

The syllabus is meant as a guideline. For a lot of the second quarter, I was behind, and (Blakey) never said anything. He also suggests homework problems, but I never look at those. Other teachers and I are doing things differently in our classes, and he doesn't mind. So, I'm surprised that people would think that this is being jammed down people’s throats.

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Final Thoughts

Despite these issues related to management style, the department is dealing with

the challenges positively. They are working constructively with the course materials, and

they are collaborating to a degree with the leadership and each other. I argue that other

departments under similar pressures might react in a far more negative manner, putting

such reform efforts in jeopardy. As Fullan and Stiegelbauer (1991) write,

Innovation can be a two-edged sword. It can either aggravate teachers' problems or provide a glimmer of hope. It can worsen the conditions of teaching however unintentionally, or it can provide the support, stimulation and pressure to improve. (p. 126)

Perhaps at this relatively early stage of the implementation process, one

might argue that the department has felt both edges of this “two-edged” sword. At

times, the teachers have certainly been aggravated with the magnitude and the

pace of the changes, and this uncertainty has caused stressful conditions. Yet, the

department generally seems to see progress, and believes in the direction they are

moving. This is consistent with observations from Waugh and Punch (1987), who

note that meaningful change is a long process influenced by many factors, not an

event.

Evidence of this process is seen in the reflections of one teacher who

noted, "Many people at the very beginning thought that this was just (the

department chair's) ideas, and they weren’t necessarily the best ideas. But as

people are jumping on board, and really getting into this stuff, they’re finding out

that he was right. He was right about a lot of this."

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CHAPTER SIX

SUMMARY AND CONCLUSIONS

An incredibly complex process has begun in the Adderley High Schools'

mathematics departments. The new materials introduced in 1997 are slowly being phased

in at East and West Adderley, and traditions, values, practices and beliefs are

consequently being rethought and reshaped. The culture of the collective department is

changing as the staff, students and administration all adapt to a new paradigm. For

teachers in particular, with new paradigms come new occupational (and perhaps

personal) identities, and educators need to be given opportunities to construct those

identities (Norum & Lowry, 1995, p. 5). With this in mind, I examined some cultural

characteristics of the Adderley mathematics department that may have supported these

changes, which are perhaps the beginning of the construction of these new identities.

Summary

In this paper, I initially discussed some aspects of paradigm shifts among

mathematics teachers and why these changes are important if reform is to occur.

Specifically, I suggested that teachers and departments need to reexamine traditional

practices if mathematics is truly to become "for all". Such a process has begun at

Adderley High School, and I began to describe the situation. Following a brief

explanation of reform mathematics versus traditional mathematics, I discussed details of

the school environment at Adderley, including student characteristics, school climate, and

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the surrounding community. In addition, I began to describe my own history with the

department and how it influenced this study.

The second chapter examined some of the literature relevant to the characteristics

of mathematics teachers, both in general and as these characteristics relate to

mathematics reform. Specifically, this discussion noted mathematics teachers' receptivity

toward reform-oriented curricula and instructional methods, such as new definitions of

the role of the teacher and instructional activities. In addition, I noted research related to

the characteristics, culture and receptivity to change of mathematics departments as a

whole.

This research indicated that there is a wide variation in the psychological states of

teachers, and some of these characteristics may lead some teachers to be uniquely

resistant to change. Consistent with this observation, research shows that mathematics

teachers are not universally receptive or open to the types of reforms recommended by

the NCTM Standards. Three themes were identified as important to a teacher's receptivity

to reform-oriented practices and philosophies: beliefs about mathematics as a field of

study, beliefs about the teacher's role, and the ability to reflect.

For example, if teachers hold a largely absolutist view of mathematics, they may

have a more difficult time implementing or making these recommendations personally

meaningful. Furthermore, teachers that were inclined to promote an open atmosphere of

discussion and exploration in the classroom indicated greater receptivity toward

mathematics reform, and had an easier time implementing reform-oriented methods. In

contrast, holding a control-oriented concept of the teacher's role proved more problematic

in implementing these methods. In addition, the ability and opportunity to reflect on

87

teaching practices was consistently noted as an indicator of receptivity to mathematics

reform.

Departmental receptivity to reform was associated with, and affected by, several

elements. Leadership within the department that promoted practices consistent with the

NCTM Standards was significant, as was a department's overall involvement in

professional activities. This related to outside activities such as conference attendance, as

well as day-to-day interactions with colleagues. Furthermore, departments that held

constructively high expectations for all students were more successful in encouraging the

achievement of a greater number of students in higher levels of mathematics classes.

Finally, mathematics departments that were collectively committed to change were

predictably more successful in implementing change.

The third chapter describes the rationale behind the methodology of this study. It

elaborated on the unique aspects of the Adderley High Schools' mathematics curriculum,

and specific attributes of the Core-Plus and the Accelerated Mathematics programs. I

further elaborated on my history with the department, and my personal reflections on the

departmental culture. The development of this study arose from my interest in these

issues and how they would affect the implementation process at Adderley.

I explored reasons that this department may or may not have been receptive to this

curriculum change, in relation to criteria teachers use in assessing changes as discussed

by Fullan and Stiegelbauer (1991). Despite the fact that this new curriculum would

necessarily affect these teachers in terms of time and energy, and would challenge

traditionally held beliefs, there was a perception in the department that a change was

needed. This chapter then outlines the details of the two stages of data collection.

88

Chapter Four describes the findings of the first stage of data collection, which

involved a survey completed by all Core-Plus teachers at Adderley, and interviews with

nine of them. The surveys asked the teachers to gauge the extent of their agreement with

several statements related to reform-oriented mathematics teaching, and the Core-Plus

curriculum in particular. Furthermore, the teachers were asked to estimate the extent to

which they intended to use certain instructional strategies. The interviews then explored

teachers' metaphors for mathematics teaching, learning, and mathematics as a field of

study. In addition, teachers elaborated on their conceptions of authority and control in the

classroom.

The results of these data changed the direction of the overall study. Initially, the

focus of this work was to be an examination of teachers' conceptions of control and

authority, and the extent to which these affected the implementation of a student-centered

curriculum. However, I was surprised at the level of consistency between the teachers'

responses and recommendations made by national mathematics reform documents. That

is, the department seemed to be quite receptive to recommendations related to classroom

management; most of those interviewed suggested that they control classes by "letting

go". This suggested that a broader examination of the situation at Adderley was

warranted in order to better gauge the level of this receptivity and the elements that have

affected it.

The second stage of data collection consisted only of interviews which focused on

teachers' general perceptions of the new curriculum and their impressions of overall

departmental changes over time. Specifically, they were asked to reflect on leadership,

89

instructional practices, and professional development. As a result of these data, a fuller

picture of the department emerged.

In general, the department is encouraged by the results of the new curriculum,

although some tensions must still be addressed. These are: the perception that the Core-

Plus curriculum does not teach basic mathematical skills; the question of whether the

reform-oriented curriculum is appropriate for honors level students; disagreement over

the compatibility of Core-Plus and the Accelerated Mathematics Program; and tensions

between younger and veteran staff.

However, aspects of the process ultimately point toward successful

implementation. In particular, Fullan and Stiegelbauer (1991) discuss elements crucial to

successful reform efforts, such as vision building, restructuring, staff development and

resource assistance, monitoring and problem coping, evolutionary planning, and initiative

taking and empowerment. Most of these have been dealt with in ways that Fullan and

Stiegelbauer recommend, although the department chair's leadership style has been

challenged as authoritarian. This top-down style is not a universal concern among the

staff, but it should be noted as an area of concern, and may conflict with encouraging an

empowered faculty.

Conclusions

Despite these disputes, I argue that leadership has been the key element to the

implementation of the new curriculum. Not only has the upper level and departmental

leadership been crucial, but leadership by individual teachers has also taken a strong role.

There are a number of passionate advocates of the new program who are committed and

90

excited about this new direction. As one teacher noted, "We won't go back…There are

enough teachers here that nurture, that don't want (the program) to fail." Looking to the

future, these teachers see the importance of encouraging leadership roles among their

peers. One teacher suggested,

Ultimately, individual teachers will become leaders, and they'll move it forward. I don't think we'll go backwards in it (to a traditional curriculum); we'll keep modifying it, and try to bring it closer to…making everybody happy. Hopefully, eventually, the goal of the math dept will be the same as the Prairie State Test, which will be the same as the goal of the ACT test, and we'll all be on the same page and everything will be in tune. Then, we won't have to worry about those leadership issues.

Although the empowerment issue is a concern, this comment indicates that there

are those who are looking toward taking initiative and see it not only possible, but

perhaps necessary for the continued success of the program.

Looking at the entire process, it is not surprising to see that there has been pain

and frustration involved. In fact, conflict and disagreement may be inevitable and

fundamental to change processes (Fullan & Stiegelbauer, 1991). A more colorful

metaphor for this comes from Norum and Lowry (1995), who compare change to a death,

the death of traditions and familiar assumptions. Continuing this metaphor, they suggest

that educators experiencing change go through Kubler-Ross's (1969) stages of grief:

denial, anger, bargaining, depression, and finally, acceptance. Therefore, as with any

family coming to terms with such a loss, one can expect difficulties and painful issues. At

the time of this study, teachers that could be described as experiencing grief for the old

curriculum (many were not) were at different stages of this process. However, the

department as a whole appears to be moving toward acceptance.

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Furthermore, looking past the uncertainty and tensions that are common so early

in such an implementation process, it is natural to try to forecast the larger implications of

this effort. As I have noted, I see this department on the path toward successful

implementation of the Core-Plus curriculum. Whether or not it remains fused together

with the Accelerated Mathematics Program will be seen in time. However, I believe that

this department has "passed the point of no return", and as the teacher above noted, it

would be difficult for them to return to a traditional curriculum anytime soon.

Beyond the details of implementing Core-Plus or Ack Math, the Adderley

mathematics department may be involved in a more significant process. Figure 5 is a

model adapted from Fullan and Stigelbauer (1991) and Gutierrez (1996) which illustrates

the potential effects of educational leadership on reform issues, departmental culture, and

student outcomes. In Chapter Five, I argued that the departmental leadership at Adderley

High Schools has mostly addressed the six significant aspects of educational reform

presented by Fullan and Stiegelbauer (1991), and this has impacted the departmental

culture as a whole. Furthermore, in Chapter Three I indicated that I see this department as

potentially reshaping its culture as Organized For Advancement (OFA), as outlined by

Gutierrez (1996). Thus, leadership affects culture, which may ultimately lead to student

outcomes similar to those advocated by mathematics reform literature, such as the NCTM

Standards documents.

The OFA Framework

Again, OFA is a framework developed to describe the practices, beliefs and general

culture of mathematics departments effective in encouraging their students to continue

onto high levels of mathematics (Gutierrez, 1996).

92

Insert figure 5 here.

93

OFA departments specifically show a high level of overall student participation in

mathematics and a high level of student participation in upper level mathematics courses.

In contrast, non-OFA departments maintained placement policies that ensured only a

select number of students were encouraged to take additional and upper level

mathematics courses. Gutierrez's study has particular relevance here, because the schools

included in her study were very similar to Adderley. That is, each school was mainly

urban, enrolled approximately 1000-2000 total students, served a significant proportion

of low income and minority students, and were identified as schools serving students that

traditionally underperform in mathematics (Gutierrez, 1996).

The framework outlines four components common to OFA departments: a

rigorous and common curriculum, innovative instructional practices, a commitment to a

collective enterprise, and a commitment to all students. These components are explained

more fully below as they relate to Adderley High Schools.

A Rigorous and Common Curriculum

The Adderley program seems particularly consistent with this aspect of the

Organized For Advancement framework. OFA departments generally offer fewer low-

level mathematics courses, and offer a limited choice in courses overall. This

"compressed curricula" (p. 508) leads to students that take more mathematics, and also

higher levels of mathematics. Similarly, the Adderley mathematics department has

reduced the number of courses it offers, and has compressed its "tracking" lanes from

three to two. As noted, all freshman-level courses use the same text. A low-level "Pre-

Algebra" course was eliminated in 1998-1999.

94

Unfortunately, one significant area of difference between Adderley High Schools

and schools that are Organized For Advancement relates to high school graduation

requirements. All of the schools identified as OFA by Gutierrez had a three-year

mathematics requirement; Adderley currently has only a two-year requirement. Gutierrez

writes, "Porter (1994) found that when schools serving traditionally low-achieving

students increased their standards for graduation, their students took more mathematics

and at varying levels during high school" (1996, p.509). This may not be a particularly

surprising finding, but it is important to note. In addition, the fact that Core-Plus has been

designed as a three-year curriculum for all students weighs heavily on this graduation

requirement.

However, this has been discussed within the department and with upper level

administration. Because the State of Illinois plans to implement the Prairie State Test, a

standardized assessment that tests objectives through the junior year of high school, the

department plans to improve the program so that students will voluntarily take an

additional year of mathematics. This issue has not been resolved, and a three-year

mathematics requirement may be adopted at a later date.

Innovative Instructional Practices

In addition, the new mathematics curriculum discussed in this paper seems to lend

itself to this type of innovation. Although the OFA departments studied by Gutierrez still

used lecture format as the primary mode of instruction, new instructional approaches

tended to be used to supplement instruction more often than in non-OFA sites.

Specifically, these approaches are the use of cooperative learning, the use of technology,

95

the use of a variety materials, and content that related to the students' lives. These seem to

be remarkably consistent with the type of instruction encouraged by the Core-Plus/Ack

Math hybrid, as has been discussed at some length in this paper.

A Commitment to a Collective Enterprise

Here, the connection between OFA departments and Adderley is less clear.

Teachers in OFA departments tended to teach courses featuring a wide range of student

abilities and ages, rather than teaching exclusively one segment of the population (such as

the "honors" level). This had the effect of exposing teachers to the entire curriculum and

student body, and rotating course assignments in this way "helps keep you on your toes",

in the words of one teacher (Gutierrez, 1996, p. 514).

Although seniority in the department does not seem to dictate course assignments

at Adderley, there are several teachers in the department that teach only honors courses,

and have done so for years. It is unclear whether this trend will change with the full

implementation of the new curriculum. However, this practice is limited to a select few

teachers, and most others seem to teach a wide variety of students. In addition, a number

of retirements in the next several years may shake loose some of these "honors-only"

positions.

In addition, OFA departments were characterized by "cooperative teacher

autonomy" (p. 515). In other words, the teachers in these departments did not generally

interact with each other on a daily basis about instructional practices, but instead relied

more heavily on a shared conception of the department's goals to guide their work.

Because the curriculum at Adderley is so new, the teachers are still developing their

96

perceptions of the department's goals and mission. As a result, they seem to be relying on

each other for daily support much more than Gutierrez found in OFA departments.

Of course, such reliance is a natural and healthy outcome for a department

implementing a new curriculum. In fact, collegiality, communication and mutual support

among teachers have been identified as strong indicators of implementation success

(Fullan & Stiegelbauer, 1991). In time, the Adderley mathematics department may indeed

develop this level of cooperative teacher autonomy, but for now, they need each other for

support.

Furthermore, OFA departments were noted to be active in professional

development. The extended curriculum training provided by the CPMP publishers

continues to be supported and encouraged by the Adderley administration, as has been

noted. More frequent formalized professional development sessions take place at bi-

weekly "inservice" meetings, which are held in all departments. One teacher had a unique

perspective on how these meetings have changed since the implementation of the new

program, particularly as a result of the existence of a common curriculum for all teachers.

He notes, "Now, everybody has a standard….when we talk at inservice, it applies to

everybody. Consequently, there’s a lot more sharing. Now, at department meetings, I see

growth. It’s constant growth."

Finally, OFA departments practiced collective decision-making. This issue has

already been described as controversial within the department, and several conflicting

reports of the level of collective decision-making allowed by the departmental leadership

have surfaced. This is an issue to be explored further.

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A Commitment to all Students

This component is in need of the most further study as it relates to the Adderley

mathematics department. Gutierrez suggests several dimensions of a department's

commitment to all students (p. 510):

1. Teachers holding constructive conceptions of students2. Teachers holding flexible conceptions of the learning process3. Teachers sharing the responsibility for learning with students4. Teachers holding high expectations for students5. Teachers actively reaching out to parents6. Teachers being accessible to students (p. 510)

In light of my earlier perception of elitism, this seems the most problematic

component of the OFA framework for this department. However, the first stage of data

collection indicated that these teachers may hold flexible conceptions of the learning

process; a control-orientation seems to have given way to student-centered teaching. In

addition, all Adderley teachers are required to be available to their students every day in

"period 10", a 35-minute period after school in which students can approach teachers for

help in their classrooms on a walk-in basis. Also encouraging is that as of the 2001-2002

school year, the two "tracking lanes" at Adderley are labeled "College Prep" and

"Accelerated College Prep." This indicates that the leadership has made a commitment to

encouraging the attitude that all students are potential college students. Still, the findings

of this study did not adequately assess the level of commitment this staff has to all

students, as defined by Gutierrez. Again, this is an area in need of further study.

Despite these uncertainties, I still contend that the Adderley mathematics

department is a potential OFA department. Because OFA sites were identified as such

based on certain student outcomes, and the Adderley has not had time to measure these

98

outcomes, time will tell whether my hypothesis is correct. However, taken as a whole,

this department is showing signs of reshaping itself as I have argued. Again, it is early in

the implementation process, but key elements of a department that is Organized For

Advancement are in place.

Future Areas of Study

Certainly, the logical next step for future study would be to conduct classroom

observations, as all of the conclusions here are based on the highly subjective nature of

teacher interviews. Specifically, we must see if the department truly features "more

humanized classrooms", as suggested by Mr. Blakey. The perception in the department

seems to be that instructional practices are changing as teachers become more

comfortable with the curriculum. However, this cannot be taken for granted. As Fullan

and Stiegelbauer suggest, understanding reform policies and actually implementing them

are entirely different matters (1991).

In addition, it would be extremely useful to conduct interviews with students.

Because one of the claims made by the CPMP developers is that an outcome of the

curriculum is an improvement in student attitudes and perceptions of mathematics, this

should be explored at Adderley. Although the developers' claim has been substantiated by

some research (Schoen & Pritchett, 1998), it is not yet clear whether this hypothesis holds

true for this situation.

Additionally noted is the necessity of collecting data related to student outcomes.

That is, even if the teachers are implementing the curriculum as designed, and have truly

accepted a reform-oriented paradigm for teaching and learning, do the students actually

99

benefit? Standardized achievement test results, overall student enrollment in

mathematics, student enrollment in upper level mathematics courses, and student

attitudinal data should all be considered as a part of this process. The department has

indicated that this is an important aspect of the overall implementation plan, but these

data must be collected and analyzed thoroughly and effectively.

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APPENDIX A

TEACHER SURVEY

Date _________

Part I: Background information

a. Name ____________________________________________

b. Sex F M

c. Age

d. Total years of teaching experience, not including student teaching:

e. Years of experience teaching Core-Plus materials:

f. Approximately how many hours have you spent in professional development activities geared toward the Core-Plus curriculum?

g. What year(s) of Core –Plus will you be teaching this year?

h. What is your total number of teaching periods?

i. What is your total number of periods teaching Core-Plus classes?

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j. From which institutions do you do you hold post-secondary degrees?

Degree held Institution

Part II.

Express the extent of agreement between the feeling expressed in each statement and your personal feelings. Circle the choice that best reflects your feelings.

a. "Mathematics will change rapidly in the near future."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

b. "Mathematics is a good field for creative people."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

c. "Mathematics education is a good field for creative people."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

d. "I am familiar with the National Council of Teachers of Mathematics

Curriculum and Evaluation Standards."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

e. "I am familiar with the Illinois State Learning Standards."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

f. "The mathematics content of the Core-Plus curriculum is rigorous and challenging."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

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g. " As compared to other classes, my methods of presenting mathematical content will stay the same in implementing the Core-Plus Curriculum."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

h. " As compared to other classes, my classroom management strategies will stay the same in implementing the Core-Plus curriculum."

Strongly Disagree

Disagree Undecided Agree Strongly Agree

Part III. Over the duration of the Core-Plus course(s) that you teach, please indicate how often you intend to employ the following classroom teaching strategies. Circle the choice that best reflects your feelings: a) Lecture: Rarely, if ever Monthly Weekly Several times

per weekEvery day

b) Guided discovery activities:

Rarely, if ever Monthly Weekly Several times per week

Every day

c) Group work:Rarely, if ever Monthly Weekly Several times

per weekEvery day

d) Computer work:Rarely, if ever Monthly Weekly Several times

per weekEvery day

e) Individual work (seat work):

Rarely, if ever Monthly Weekly Several times per week

Every day

h) Graphing calculator work:

Rarely, if ever Monthly Weekly Several times per week

Every day

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i) Class discussion:Rarely, if ever Monthly Weekly Several times

per weekEvery day

Part IV. Over the duration of the Core-Plus course(s) that you teach, please indicate how frequently you intend to emphasize each of the following in your instruction. Circle the choice that best reflects your feelings: a) Mathematical reasoning --passive (YOU show your students justification for mathematical statements; YOU demonstrate deductive proofs of theorems)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

b) Mathematical reasoning --active (your STUDENTS construct justifications for mathematical statements; STUDENTS construct demonstrative proofs of theorems)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

c) Mathematical procedures (your students develop skill in performing routine mathematical procedures, algorithms)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

d) Mathematical facts (your students learn basic and statements of basic theorems)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

e) Applications of mathematics -- passive (YOU show your students how mathematics can be used to solve a variety of "real world" problems, e.g., in science)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

f) Applications of mathematics -- active (your STUDENTS determine how mathematics can be used to solve a variety of "real world" problems, e.g., in science)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

g) Mathematical communication (you ask your students to speak and/or write clearly about mathematical ideas)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

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h) Modeling --passive (YOU show your students how to create and use appropriate mathematical representations of "real world" situations, e.g., in science)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

i) Modeling --active (your STUDENTS create and use their own appropriate mathematical representations of "real world" situations, e.g., in science)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

j) Multiple representations (your students see and work with the same mathematical concept in a variety forms; graphical, numerical [tabular], symbolic, and verbal)Not emphasized at all

Occasionally, but not frequently

Emphasized frequently

Almost always

Part V. Please indicate with a check your level of expertise in using the following resources/strategies in your teaching. Circle the choice that best reflects your feelings:

a) Classroom Lectures/Presentations:

Novice Intermediate Expert

b) Group Work:Novice Intermediate Expert

c) Extended Projects:Novice Intermediate Expert

d) Writing Activities:Novice Intermediate Expert

e) Computers:Novice Intermediate Expert

f) Graphing Calculators:Novice Intermediate Expert

g) Portfolios and other types of non-routine assessments:

Novice Intermediate Expert

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Interview #1

Name____________________________

Date __________________________

1. Select the three of the choices below that best characterize your beliefs about

mathematics teaching. Please explain to me why you made these choices.

“Teaching mathematics is most like being a…”

News broadcaster Entertainer Doctor Orchestra conductor Gardener Coach Missionary Social worker Political leader Other: ____________________________

2. Select the three of the choices below that best characterize your beliefs about mathematics learning. Please explain to me why you made these choices.

“Learning mathematics is most like…”

Working on an assembly line Watching a movie Cooking with a recipe Picking fruit from a tree Working a jigsaw puzzle Conducting an experiment Building a house Creating a clay sculpture A negotiation Other:_______________________________

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3. Select the three of the choices below that best characterize your beliefs about mathematics as a field of study. Please explain to me why you made these choices.

“Mathematics is most like….”

A set of laws A blueprint A piece of art A puzzle A cookbook A language A philosophy A frontier Other: ____________________________

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APPENDIX B

CORE -PLUS TEXTBOOK SAMPLE

INSERT REPRODUCTIONS HERE

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APPENDIX C

ACCELERATED MATHEMATICS PROGRAM SAMPLE QUESTIONS

INSERT QUESTIONS HERE

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APPENDIX D

COURSE SELECTIONS AND CURRICULUM STRUCTURE

Course selections, 1999-2000

Grouping Freshman Sophomore Junior Senior

S-Group Integrated Algebra I

Integrated Geometry

Integrated Geometry

Integrated Algebra II

Regular Geometry Algebra II College AlgebraHonors Integrated

Geometry Accelerated

Geom/Chem Honors Plus or Geom. Honors

Trig/PreCalcor Algebra II Honors

AP Calculus or Trig/Precalc

Course selections, 2000-2001

Grouping Freshman Sophomore Junior Senior

S-Group Integrated Algebra I

Integrated Geometry Integrated Algebra II

Integrated Algebra II

Regular Algebra II Integrated College Algebra

Honors Integrated Geometry Accelerated

Integrated Algebra II Honors Plus or Integrated Algebra II Accelerated

Trig/PreCalcor Algebra II Honors

AP Calculus or Trig/Precalc

Course selections, 2001-2002

Grouping Freshman Sophomore Junior Senior

College Prep Integrated Algebra I

Integrated Geometry Integrated Algebra II

Integrated College with an option to take AP Statistics

Accelerated College Prep

Integrated Geometry Accelerated

Integrated Algebra II Honors Plus or Integrated Algebra II Accelerated

Integrated Trig/PreCalc Acceleratedor Integrated College Algebra Accelerated

AP Calculus and/or AP Statistics

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APPENDIX E

SAMPLE ADDERLEY MATHEMATICS DEPARTMENT SYLLABUS

Insert syllabus here

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